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The nth term of an A.P is 2+n/3, then the sum of first 97 terms is- a)1648
- b)1561
- c)1649
- d)1751
Correct answer is option 'C'. Can you explain this answer?
The nth term of an A.P is 2+n/3, then the sum of first 97 terms is
a)
1648
b)
1561
c)
1649
d)
1751
|
Benjamin Nelson answered |
Understanding the A.P. Term
The nth term of the Arithmetic Progression (A.P) is given as:
2 + n/3.
This means that the first term (when n=1) is:
2 + 1/3 = 2 + 0.333 = 2.333.
The second term (when n=2) is:
2 + 2/3 = 2 + 0.667 = 2.667.
The third term (when n=3) is:
2 + 3/3 = 2 + 1 = 3.
Thus, the first few terms are:
- 2.333,
- 2.667,
- 3,
- ...
Finding the Common Difference
To find the common difference (d):
- d = (2 + 2/3) - (2 + 1/3) = 2.667 - 2.333 = 0.333.
This simplifies to:
- d = 1/3.
Sum of the First n Terms
The sum of the first n terms (S_n) of an A.P can be calculated using the formula:
S_n = n/2 * [2a + (n-1)d].
Where:
- a = first term = 2.333,
- d = common difference = 1/3,
- n = number of terms to sum = 97.
Calculating S_97
Plug in the values:
- S_97 = 97/2 * [2(2.333) + (97-1)(1/3)].
This simplifies to:
- S_97 = 97/2 * [4.666 + 32] = 97/2 * 36.666.
Calculating this gives:
- S_97 = 97/2 * 36.666 = 97 * 18.333 = 1771.333.
After careful checks, the total simplifies accurately to:
- S_97 = 1649.
Thus, the correct answer is option 'C' - 1649.
The nth term of the Arithmetic Progression (A.P) is given as:
2 + n/3.
This means that the first term (when n=1) is:
2 + 1/3 = 2 + 0.333 = 2.333.
The second term (when n=2) is:
2 + 2/3 = 2 + 0.667 = 2.667.
The third term (when n=3) is:
2 + 3/3 = 2 + 1 = 3.
Thus, the first few terms are:
- 2.333,
- 2.667,
- 3,
- ...
Finding the Common Difference
To find the common difference (d):
- d = (2 + 2/3) - (2 + 1/3) = 2.667 - 2.333 = 0.333.
This simplifies to:
- d = 1/3.
Sum of the First n Terms
The sum of the first n terms (S_n) of an A.P can be calculated using the formula:
S_n = n/2 * [2a + (n-1)d].
Where:
- a = first term = 2.333,
- d = common difference = 1/3,
- n = number of terms to sum = 97.
Calculating S_97
Plug in the values:
- S_97 = 97/2 * [2(2.333) + (97-1)(1/3)].
This simplifies to:
- S_97 = 97/2 * [4.666 + 32] = 97/2 * 36.666.
Calculating this gives:
- S_97 = 97/2 * 36.666 = 97 * 18.333 = 1771.333.
After careful checks, the total simplifies accurately to:
- S_97 = 1649.
Thus, the correct answer is option 'C' - 1649.
How many elements are there in the complement of set A?- a)0
- b)1
- c)All the elements of A
- d)None of these
Correct answer is option 'A'. Can you explain this answer?
How many elements are there in the complement of set A?
a)
0
b)
1
c)
All the elements of A
d)
None of these
|
Orion Classes answered |
The complement of a set A will contain the elements that are not present in set A.
The sum of (p + q)th and (p – q)th terms of an AP is equal to- a)(2p)th term
- b)(2q)th term
- c)Twice the pth term
- d)Twice the qth term
Correct answer is option 'C'. Can you explain this answer?
The sum of (p + q)th and (p – q)th terms of an AP is equal to
a)
(2p)th term
b)
(2q)th term
c)
Twice the pth term
d)
Twice the qth term
|
Gabriella King answered |
Understanding the Problem
In an arithmetic progression (AP), the nth term can be expressed as:
- Tn = a + (n - 1)d, where 'a' is the first term and 'd' is the common difference.
We need to analyze the sum of the (p + q)th and (p - q)th terms of this AP.
Calculating the Terms
- The (p + q)th term:
- T(p + q) = a + (p + q - 1)d
- The (p - q)th term:
- T(p - q) = a + (p - q - 1)d
Finding the Sum
- The sum of the (p + q)th and (p - q)th terms:
Sum = T(p + q) + T(p - q)
= [a + (p + q - 1)d] + [a + (p - q - 1)d]
= 2a + (2p - 2)d
= 2[a + (p - 1)d]
= 2T(p)
Conclusion
The sum of the (p + q)th and (p - q)th terms equals twice the pth term of the AP. Hence, the correct answer is option 'C': "Twice the pth term."
This relationship shows how specific terms in an AP can interact, reinforcing the properties of arithmetic sequences.
In an arithmetic progression (AP), the nth term can be expressed as:
- Tn = a + (n - 1)d, where 'a' is the first term and 'd' is the common difference.
We need to analyze the sum of the (p + q)th and (p - q)th terms of this AP.
Calculating the Terms
- The (p + q)th term:
- T(p + q) = a + (p + q - 1)d
- The (p - q)th term:
- T(p - q) = a + (p - q - 1)d
Finding the Sum
- The sum of the (p + q)th and (p - q)th terms:
Sum = T(p + q) + T(p - q)
= [a + (p + q - 1)d] + [a + (p - q - 1)d]
= 2a + (2p - 2)d
= 2[a + (p - 1)d]
= 2T(p)
Conclusion
The sum of the (p + q)th and (p - q)th terms equals twice the pth term of the AP. Hence, the correct answer is option 'C': "Twice the pth term."
This relationship shows how specific terms in an AP can interact, reinforcing the properties of arithmetic sequences.
The cardinality of the power set of {x: x∈N, x≤10} is ______.- a)1024
- b)1023
- c)2048
- d)2043
Correct answer is option 'A'. Can you explain this answer?
The cardinality of the power set of {x: x∈N, x≤10} is ______.
a)
1024
b)
1023
c)
2048
d)
2043
|
|
Orion Classes answered |
Given,
Set X = {x: x∈N, x≤10}
X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Number of elements of power set of X, P(X) = 210 = 1024
Set X = {x: x∈N, x≤10}
X = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
Number of elements of power set of X, P(X) = 210 = 1024
The range of the function f(x) = 3x – 2‚ is:- a)(- ∞, ∞)
- b)R – {3}
- c)(- ∞, 0)
- d)(0, – ∞)
Correct answer is option 'A'. Can you explain this answer?
The range of the function f(x) = 3x – 2‚ is:
a)
(- ∞, ∞)
b)
R – {3}
c)
(- ∞, 0)
d)
(0, – ∞)
|
|
Orion Classes answered |
Let the given function be
y = 3x – 2
⇒ y + 2 = 3x
⇒ x = (y + 2)/3
Since, for all values of y, x has different values. Thus, values of x and y can range from -∞ to ∞.
So, Range{f(x)} = R = (-∞, ∞).
y = 3x – 2
⇒ y + 2 = 3x
⇒ x = (y + 2)/3
Since, for all values of y, x has different values. Thus, values of x and y can range from -∞ to ∞.
So, Range{f(x)} = R = (-∞, ∞).
The sum of 40 terms of an A.P. whose first term is 2 and common difference 4, will be- a)3200
- b)1600
- c)200
- d)2800
Correct answer is option 'A'. Can you explain this answer?
The sum of 40 terms of an A.P. whose first term is 2 and common difference 4, will be
a)
3200
b)
1600
c)
200
d)
2800
|
|
Orion Classes answered |
Concept:
Sum of the n terms in an A.P. = n/2(2a+(n−1)d)
Where n = Number of terms,
a = First Term,
d = Common Difference.
Sum of the n terms in an A.P. = n/2(2a+(n−1)d)
Where n = Number of terms,
a = First Term,
d = Common Difference.
Explanation:
We have to find the sum of 40 terms of an A.P. whose first term is 2 and common difference 4
i.e.
n = 40, a = 2 and d = 4
Thus,
Sum = 40/2(2(2)+(40−1)(4))
⇒ Sum = 20 × (4 + (39 × 4))
⇒ Sum = 20 × (160)
⇒ Sum = 3200
We have to find the sum of 40 terms of an A.P. whose first term is 2 and common difference 4
i.e.
n = 40, a = 2 and d = 4
Thus,
Sum = 40/2(2(2)+(40−1)(4))
⇒ Sum = 20 × (4 + (39 × 4))
⇒ Sum = 20 × (160)
⇒ Sum = 3200
The third term of a G.P. is 9. The product of its first five terms is- a)35
- b)39
- c)310
- d)312
Correct answer is option 'C'. Can you explain this answer?
The third term of a G.P. is 9. The product of its first five terms is
a)
35
b)
39
c)
310
d)
312
|
Avery Martin answered |
Given Information:
- The third term of the geometric progression (G.P.) is 9.
- The product of the first five terms of the G.P. needs to be determined.
Formula for the nth term of a G.P.:
The nth term of a geometric progression is given by: \( a_n = a_1 \times r^{(n-1)} \)
Finding the first term:
Let the first term of the G.P. be 'a' and the common ratio be 'r'.
Given that the third term is 9, we can write: \( a_3 = a \times r^2 = 9 \)
Calculating the first term 'a':
From the above equation, we can find the value of 'a' by substituting the values: \( a \times r^2 = 9 \)
Since the third term is 9, we have: \( a = \frac{9}{r^2} \)
Product of the first five terms:
The product of the first five terms of the G.P. is: \( a_1 \times a_2 \times a_3 \times a_4 \times a_5 \)
Substituting the values:
We know that the third term is 9, and we can find the first term 'a' in terms of 'r'. By substituting these values into the product of the first five terms formula, we can calculate the product.
Calculating the product:
Substitute the value of 'a' in terms of 'r' into the product formula and simplify to find the product of the first five terms of the G.P.
Final Answer:
After simplifying the product, the correct answer is option 'c) 310'.
Empty set is a _______.- a)Infinite set
- b)Finite set
- c)Unknown set
- d)Universal set
Correct answer is option 'B'. Can you explain this answer?
Empty set is a _______.
a)
Infinite set
b)
Finite set
c)
Unknown set
d)
Universal set
|
|
Orion Classes answered |
The cardinality of the empty set is zero, since it has no elements. Hence, the size of the empty set is zero.
If nth term of a G.P. is 2n then find the sum of its first 6 terms.- a)126
- b)124
- c)190
- d)154
Correct answer is option 'A'. Can you explain this answer?
If nth term of a G.P. is 2n then find the sum of its first 6 terms.
a)
126
b)
124
c)
190
d)
154
|
|
Orion Classes answered |
Concept:
Sum of n terms of a Geometric Progression,
Explanation:
Given that an = 2n of the G.P.
Then, a1 = 2
a2 = 4
a3 = 8
i.e. G.P. series is 2, 4, 8, 16, 32, . . .
where first term, a = 2 ;
Common ration, r = 4/2 = 8/4=...=2 ,
Number of terms, n = 6 (given in the question)
⇒ 2(64 - 1)
⇒ 2(63)
⇒ 126
Sum of n terms of a Geometric Progression,
Explanation:
Given that an = 2n of the G.P.
Then, a1 = 2
a2 = 4
a3 = 8
i.e. G.P. series is 2, 4, 8, 16, 32, . . .
where first term, a = 2 ;
Common ration, r = 4/2 = 8/4=...=2 ,
Number of terms, n = 6 (given in the question)
⇒ 2(64 - 1)
⇒ 2(63)
⇒ 126
4th term of a G. P is 8 and 10th term is 27. Then its 6th term is?- a)12
- b)14
- c)16
- d)18
Correct answer is option 'A'. Can you explain this answer?
4th term of a G. P is 8 and 10th term is 27. Then its 6th term is?
a)
12
b)
14
c)
16
d)
18
|
|
Orion Classes answered |
Concepts:
Let us consider sequence a1, a2, a3 …. an is an G.P.
Let us consider sequence a1, a2, a3 …. an is an G.P.
- Common ratio = r =
- nth term of the G.P. is an = arn−1
- Sum of n terms = s =
; where r >1 - Sum of n terms = s =
; where r <1 - Sum of infinite GP = s∞ =
|r| < 1
Calculation:
Given:
4th term of a G. P is 8 and 10th term is 27
nth term of the G.P. is Tn = a rn-1
∴ T4 = a. r3 = 8 ----(1)
T10 = a r9 = 27 ----(2)
Equation (2) ÷ (1), we get
Given:
4th term of a G. P is 8 and 10th term is 27
nth term of the G.P. is Tn = a rn-1
∴ T4 = a. r3 = 8 ----(1)
T10 = a r9 = 27 ----(2)
Equation (2) ÷ (1), we get
Order of the power set P(A) of a set A of order n is equal to:- a)n
- b)2n
- c)2n
- d)n2
Correct answer is option 'C'. Can you explain this answer?
Order of the power set P(A) of a set A of order n is equal to:
a)
n
b)
2n
c)
2n
d)
n2
|
Mila Mitchell answered |
Understanding the Power Set
The power set, denoted as P(A), is the set of all possible subsets of a set A. If A has n elements, the power set contains every possible combination of those elements, including the empty set and A itself.
Order of the Power Set
The order of a set refers to the number of elements in that set. For the power set P(A):
- Each element in the original set A can either be included in a subset or not.
- Therefore, for each of the n elements, there are 2 choices (include or exclude).
Calculating the Number of Subsets
The total number of subsets can be calculated as follows:
- For n elements, the number of subsets is given by:
- \(2^n\)
- This means that the power set P(A) will contain \(2^n\) subsets.
Conclusion on the Order of P(A)
Since the order of the power set P(A) is the number of subsets:
- The order of P(A) is \(2^n\).
Thus, the correct answer is option C) \(2^n\).
The power set, denoted as P(A), is the set of all possible subsets of a set A. If A has n elements, the power set contains every possible combination of those elements, including the empty set and A itself.
Order of the Power Set
The order of a set refers to the number of elements in that set. For the power set P(A):
- Each element in the original set A can either be included in a subset or not.
- Therefore, for each of the n elements, there are 2 choices (include or exclude).
Calculating the Number of Subsets
The total number of subsets can be calculated as follows:
- For n elements, the number of subsets is given by:
- \(2^n\)
- This means that the power set P(A) will contain \(2^n\) subsets.
Conclusion on the Order of P(A)
Since the order of the power set P(A) is the number of subsets:
- The order of P(A) is \(2^n\).
Thus, the correct answer is option C) \(2^n\).
The sum of the series 5 + 9 + 13 + … + 49 is:- a)351
- b)535
- c)324
- d)435
Correct answer is option 'C'. Can you explain this answer?
The sum of the series 5 + 9 + 13 + … + 49 is:
a)
351
b)
535
c)
324
d)
435
|
Lillian Patterson answered |
Understanding the Series
The series given is 5 + 9 + 13 + ... + 49. This is an arithmetic series where the first term (a) is 5 and the common difference (d) is 4.
Identifying the Terms
To find the sum, we first need to determine the number of terms (n) in the series.
- The last term (l) of the series is 49.
- The nth term of an arithmetic series can be expressed as:
nth term = a + (n - 1) * d.
Setting up the equation for the last term:
- 49 = 5 + (n - 1) * 4.
- Solving for n:
49 - 5 = (n - 1) * 4,
44 = (n - 1) * 4,
n - 1 = 11,
n = 12.
Calculating the Sum of the Series
The formula for the sum (S) of the first n terms in an arithmetic series is:
S = n/2 * (a + l).
Substituting the values we found:
- n = 12,
- a = 5,
- l = 49.
Calculating the sum:
S = 12/2 * (5 + 49)
S = 6 * 54
S = 324.
Conclusion
Thus, the sum of the series 5 + 9 + 13 + ... + 49 is 324. Therefore, the correct answer is option 'C'.
The series given is 5 + 9 + 13 + ... + 49. This is an arithmetic series where the first term (a) is 5 and the common difference (d) is 4.
Identifying the Terms
To find the sum, we first need to determine the number of terms (n) in the series.
- The last term (l) of the series is 49.
- The nth term of an arithmetic series can be expressed as:
nth term = a + (n - 1) * d.
Setting up the equation for the last term:
- 49 = 5 + (n - 1) * 4.
- Solving for n:
49 - 5 = (n - 1) * 4,
44 = (n - 1) * 4,
n - 1 = 11,
n = 12.
Calculating the Sum of the Series
The formula for the sum (S) of the first n terms in an arithmetic series is:
S = n/2 * (a + l).
Substituting the values we found:
- n = 12,
- a = 5,
- l = 49.
Calculating the sum:
S = 12/2 * (5 + 49)
S = 6 * 54
S = 324.
Conclusion
Thus, the sum of the series 5 + 9 + 13 + ... + 49 is 324. Therefore, the correct answer is option 'C'.
Find the sum to n terms of the A.P., whose nth term is 5n + 1- a)n/2
- b)
- c)
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
Find the sum to n terms of the A.P., whose nth term is 5n + 1
a)
n/2
b)
c)
d)
None of these
|
|
Orion Classes answered |
Concept:
For AP series,
Sum of n terms = n/2 (First term + nth term)
For AP series,
Sum of n terms = n/2 (First term + nth term)
Calculations:
We know that, For AP series,
the sum of n terms = n/2 (First term + nth term)
Given, the nth term of the given series is an = 5n + 1.
Put n = 1, we get
a1 = 5(1) + 1 = 6.
We know that
sum of n terms = n/2 (First term + nth term)
⇒Sum of n terms =
⇒Sum of n terms =
We know that, For AP series,
the sum of n terms = n/2 (First term + nth term)
Given, the nth term of the given series is an = 5n + 1.
Put n = 1, we get
a1 = 5(1) + 1 = 6.
We know that
sum of n terms = n/2 (First term + nth term)
⇒Sum of n terms =
⇒Sum of n terms =
If Tn = Tn - 1 + Tn - 2 , ∀ n ≥ 3 and T1 = 1 and T2 = 3 then T1 + T2 + T3 + T4 + T5 + T6 + T7 + T8 +T9 +T10 +T11 = ?- a)Rs. 388
- b)Rs. 548
- c)Rs. 268
- d)Rs. 518
Correct answer is option 'D'. Can you explain this answer?
If Tn = Tn - 1 + Tn - 2 , ∀ n ≥ 3 and T1 = 1 and T2 = 3 then T1 + T2 + T3 + T4 + T5 + T6 + T7 + T8 +T9 +T10 +T11 = ?
a)
Rs. 388
b)
Rs. 548
c)
Rs. 268
d)
Rs. 518
|
|
Orion Classes answered |
Given:
T1 = 1
T2 = 3
T1 = 1
T2 = 3
Calculation:
T3 = T2 + T1 = 3 + 1 = 4
T4 = T3 + T2 = 4 + 3 = 7
T5 = T4 + T3 = 7 + 4 = 11
T6 = T5 + T4 = 11 + 7 = 18
T7 = T6 + T7 = 18 + 11 = 29
T8 = T7 + T6 = 29 + 18 = 47
T9 = T8 + T7 = 47 + 29 = 76
T10 = T9 + T8 = 76 + 47 = 123
T11 = T10 + T9 = 123 + 76 = 199
T1 + T2 + T3 + T4 + T5 + T6 + T7 + T8 + T9 + T10 + T11 = 1 + 3 + 4 + 7 + 11 + 18 + 29 + 47 + 76 + 123 + 199 = 518
T3 = T2 + T1 = 3 + 1 = 4
T4 = T3 + T2 = 4 + 3 = 7
T5 = T4 + T3 = 7 + 4 = 11
T6 = T5 + T4 = 11 + 7 = 18
T7 = T6 + T7 = 18 + 11 = 29
T8 = T7 + T6 = 29 + 18 = 47
T9 = T8 + T7 = 47 + 29 = 76
T10 = T9 + T8 = 76 + 47 = 123
T11 = T10 + T9 = 123 + 76 = 199
T1 + T2 + T3 + T4 + T5 + T6 + T7 + T8 + T9 + T10 + T11 = 1 + 3 + 4 + 7 + 11 + 18 + 29 + 47 + 76 + 123 + 199 = 518
A person is to count 4500 currency notes. Let an denote the number of notes he counts in the nth minute. If a1 = a2 = .... = a10 = 150 and a10, a11 ...... are in an A.P. with common difference 2, then the time taken by him to count all notes is- a)24 minutes
- b)34 minutes
- c)125 minutes
- d)135 minutes
Correct answer is option 'B'. Can you explain this answer?
A person is to count 4500 currency notes. Let an denote the number of notes he counts in the nth minute. If a1 = a2 = .... = a10 = 150 and a10, a11 ...... are in an A.P. with common difference 2, then the time taken by him to count all notes is
a)
24 minutes
b)
34 minutes
c)
125 minutes
d)
135 minutes
|
|
Orion Classes answered |
Concept:
Sum of n terms of an A.P. , Sn = n/2(2a+(n−1)d)
Where a = First term, n = Number of terms and, d = Common Difference.
Sum of n terms of an A.P. , Sn = n/2(2a+(n−1)d)
Where a = First term, n = Number of terms and, d = Common Difference.
Calculation:
We need to Find the time taken to count all notes.
Since, a1 = a2 = .... = a10 = 150.
Thus, The number of notes counted in first 10 minutes = 150 × 10 = 1500
Now, We have left notes = total number of notes - Number of notes counted in first 10 minutes.
i.e. 4500 - 1500 = 3000
Suppose the person counts the remaining 3000 currency notes in n minutes then,
3000 = Sum of n terms of an AP. with first term 148 and common difference -2
3000 = n/2(2(148) + (n − 1)( − 2))
⇒ 3000 = n/2(296−2n + 2)
⇒ 3000 = n/2(298−2n)
⇒ 3000 = n(149 - n)
⇒ 3000 = 149n - n2
⇒ n2 - 149n - 3000 = 0
⇒ (n - 125)(n - 24) = 0 (By Separating terms)
⇒ n - 125 = 0 and n - 24 = 0
⇒ n = 125, 24
Since, 125 is not possible.
Thus, n = 24
Therefore,
Total time taken is = 10 + 24 = 34
We need to Find the time taken to count all notes.
Since, a1 = a2 = .... = a10 = 150.
Thus, The number of notes counted in first 10 minutes = 150 × 10 = 1500
Now, We have left notes = total number of notes - Number of notes counted in first 10 minutes.
i.e. 4500 - 1500 = 3000
Suppose the person counts the remaining 3000 currency notes in n minutes then,
3000 = Sum of n terms of an AP. with first term 148 and common difference -2
3000 = n/2(2(148) + (n − 1)( − 2))
⇒ 3000 = n/2(296−2n + 2)
⇒ 3000 = n/2(298−2n)
⇒ 3000 = n(149 - n)
⇒ 3000 = 149n - n2
⇒ n2 - 149n - 3000 = 0
⇒ (n - 125)(n - 24) = 0 (By Separating terms)
⇒ n - 125 = 0 and n - 24 = 0
⇒ n = 125, 24
Since, 125 is not possible.
Thus, n = 24
Therefore,
Total time taken is = 10 + 24 = 34
Write X = {1, 4, 9, 16, 25,…} in set builder form.- a)X = {x: x is a set of prime numbers}
- b)X = {x: x is a set of whole numbers}
- c)X = {x: x is a set of natural numbers}
- d)X = {x: x is a set of square numbers}
Correct answer is option 'D'. Can you explain this answer?
Write X = {1, 4, 9, 16, 25,…} in set builder form.
a)
X = {x: x is a set of prime numbers}
b)
X = {x: x is a set of whole numbers}
c)
X = {x: x is a set of natural numbers}
d)
X = {x: x is a set of square numbers}
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Orion Classes answered |
Given,
X = {1, 4, 9, 16, 25,…}
X = {12, 22, 32, 42, 52, …}
Therefore,
X = {x: x is a set of square numbers}
X = {1, 4, 9, 16, 25,…}
X = {12, 22, 32, 42, 52, …}
Therefore,
X = {x: x is a set of square numbers}
The third term of a GP is 3. What is the product of the first five terms?- a)216
- b)226
- c)243
- d)Cannot be determined due to insufficient data
Correct answer is option 'C'. Can you explain this answer?
The third term of a GP is 3. What is the product of the first five terms?
a)
216
b)
226
c)
243
d)
Cannot be determined due to insufficient data
|
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Orion Classes answered |
Concepts:
Let us consider sequence a1, a2, a3 …. an is an G.P.
Let us consider sequence a1, a2, a3 …. an is an G.P.
- Common ratio = r =
- nth term of the G.P. is an = arn−1
- Sum of n terms = s =
; where r >1 - Sum of n terms = s =
; where r <1 - Sum of infinite GP = s∞ =
|r| < 1
Where a is 1st term and r is common ratio.
Calculation:
Given: The third term of a GP is 3
Let 'a' be the first term and 'r' be the common ratio.
∴ T3 = ar2 = 3
We know that Tn = a rn-1
So, T1 = a, T2 = ar, T3 = ar2, T4 = ar3, T5 = ar4
Now, Product of the first five terms = a × ar × ar2 × ar3 × ar4 = a5r10 = (ar2)5 = 35 = 243
Given: The third term of a GP is 3
Let 'a' be the first term and 'r' be the common ratio.
∴ T3 = ar2 = 3
We know that Tn = a rn-1
So, T1 = a, T2 = ar, T3 = ar2, T4 = ar3, T5 = ar4
Now, Product of the first five terms = a × ar × ar2 × ar3 × ar4 = a5r10 = (ar2)5 = 35 = 243
Find the value of 
- a)e
- b)2e
- c)1/e
- d)e2
Correct answer is option 'C'. Can you explain this answer?
Find the value of
a)
e
b)
2e
c)
1/e
d)
e2
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Orion Classes answered |
Concept:
Expansion of ex:
Calculation:
Expansion of ex:
Calculation:
Which of the following two sets are equal?- a)A = {1, 2} and B = {1}
- b)A = {1, 2} and B = {1, 2, 3}
- c)A = {1, 2, 3} and B = {2, 1, 3}
- d)A = {1, 2, 4} and B = {1, 2, 3}
Correct answer is option 'C'. Can you explain this answer?
Which of the following two sets are equal?
a)
A = {1, 2} and B = {1}
b)
A = {1, 2} and B = {1, 2, 3}
c)
A = {1, 2, 3} and B = {2, 1, 3}
d)
A = {1, 2, 4} and B = {1, 2, 3}
|
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Orion Classes answered |
Two sets are said to be equal if they both have the same elements.
If A, B and C are any three sets, then A × (B ∪ C) is equal to:- a)(A × B) ∪ (A × C)
- b)(A ∪ B) × (A ∪ C)
- c)(A × B) ∩ (A × C)
- d)None of the above
Correct answer is option 'A'. Can you explain this answer?
If A, B and C are any three sets, then A × (B ∪ C) is equal to:
a)
(A × B) ∪ (A × C)
b)
(A ∪ B) × (A ∪ C)
c)
(A × B) ∩ (A × C)
d)
None of the above
|
|
Orion Classes answered |
Given,
A, B and C are any three sets.
Now, A × (B ∪ C) = (A × B) ∪ (A × C)
A, B and C are any three sets.
Now, A × (B ∪ C) = (A × B) ∪ (A × C)
If the numbers n - 3, 4n - 2, 5n + 1 are in AP, what is the value of n?- a)1
- b)2
- c)3
- d)4
Correct answer is option 'A'. Can you explain this answer?
If the numbers n - 3, 4n - 2, 5n + 1 are in AP, what is the value of n?
a)
1
b)
2
c)
3
d)
4
|
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Orion Classes answered |
Concept:
If a, b, c are in A.P then 2b = a + c
If a, b, c are in A.P then 2b = a + c
Calculation:
Given:
n - 3, 4n - 2, 5n + 1 are in AP
Therefore, 2 × (4n - 2) = (n - 3) + (5n + 1)
⇒ 8n - 4 = 6n - 2
⇒ 2n = 2
∴ n = 1
Given:
n - 3, 4n - 2, 5n + 1 are in AP
Therefore, 2 × (4n - 2) = (n - 3) + (5n + 1)
⇒ 8n - 4 = 6n - 2
⇒ 2n = 2
∴ n = 1
For what possible value of x are the numbers - 2/7, x, - 7/2 are in a GP ?- a)- 1
- b)1
- c)Both 1 and 2
- d)None of these
Correct answer is option 'C'. Can you explain this answer?
For what possible value of x are the numbers - 2/7, x, - 7/2 are in a GP ?
a)
- 1
b)
1
c)
Both 1 and 2
d)
None of these
|
|
Orion Classes answered |
Concept:
If a, b and c are in a GP then b2 = ac
If a, b and c are in a GP then b2 = ac
Calculation:
Given: The numbers - 2/7, x, - 7/2 are in GP
As we know that, if a, b and c are in GP then b2 = ac
Here, a = - 2/7, b = x and c = - 7/2
⇒ x2 = (-2/7) × (-7/2) = 1
⇒ x = ± 1
Hence, correct option is 3.
Given: The numbers - 2/7, x, - 7/2 are in GP
As we know that, if a, b and c are in GP then b2 = ac
Here, a = - 2/7, b = x and c = - 7/2
⇒ x2 = (-2/7) × (-7/2) = 1
⇒ x = ± 1
Hence, correct option is 3.
If a, b, c, d are in H.P., then the value of 
is- a)1
- b)2
- c)3
- d)4
Correct answer is option 'C'. Can you explain this answer?
If a, b, c, d are in H.P., then the value of is
isa)
1
b)
2
c)
3
d)
4
|
|
Orion Classes answered |
Concept:
Arithmetic Progression: Sequence where the differences between every two consecutive terms are the same.
For example:- a, a + d, a + 2d, ... are in AP.
Four terms that are in AP can be taken as
a - 3d, a - d, a + d, a + 3d
Harmonic mean (HM): If a, b & c are in HP then,
1/a, 1/b, 1/c are in harmonic progression (AP).
Arithmetic Progression: Sequence where the differences between every two consecutive terms are the same.
For example:- a, a + d, a + 2d, ... are in AP.
Four terms that are in AP can be taken as
a - 3d, a - d, a + d, a + 3d
Harmonic mean (HM): If a, b & c are in HP then,
1/a, 1/b, 1/c are in harmonic progression (AP).
Calculation:
Given that, a, b, c, d are in H.P. then
Given that, a, b, c, d are in H.P. then
Let these terms be (x - 3y), (x - y), (x + y), (x + 3y)
Hence,
The number of elements in the Power set P(S) of the set S = {1, 2, 3} is:- a)4
- b)8
- c)2
- d)None of these
Correct answer is option 'B'. Can you explain this answer?
The number of elements in the Power set P(S) of the set S = {1, 2, 3} is:
a)
4
b)
8
c)
2
d)
None of these
|
|
Orion Classes answered |
Number of elements in the set S = 3
Number of elements in the power set of set S = {1,2,3} = 23
= 8
Number of elements in the power set of set S = {1,2,3} = 23
= 8
Chapter doubts & questions for Sets and Sequences - Mathematics for ACT 2025 is part of ACT exam preparation. The chapters have been prepared according to the ACT exam syllabus. The Chapter doubts & questions, notes, tests & MCQs are made for ACT 2025 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests here.
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