Q1:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): 184 is the 50th term of the sequence 3, 7, 11, …………
Reason (R): The nth term of A.P. is given by an = a + (n – 1)d
Explanation
let us check the position of 184 in the sequence,
an = a + (n – 1)d
184 = 3 +(n – 1)4
[a = 3 and d = 7– 3 = 4]
181 = (n – 1)4
45.25 = (n – 1)
The value of n can’t be negative.
Therefore, A is false but R is true
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Q2:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): Common difference of the A.P: -5,-1, 3, 7,……. is 4.
Reason (R): Common difference of the A.P. a, a + d, a + 2d, …….., is given by d = 2nd term – 1st term.
Explanation
Common difference is the difference between two successive terms in A.P.
d = 2nd term – 1st term = – 1 – (– 5) = –1 + 5 = 4 Therefore, Both A and R are true and R is the correct explanation for A.
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Q3:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): Sum of first hundred even natural numbers divisible by 5 is 500.
Reason (R): Sum of first n terms of an A.P is given l is last term
Explanation
First hundred even natural numbers divisible by 5 are
10, 20, 30, 40, 50............1000
If last term is 1000, then sum of all these numbers can’t be 500
Therefore, A is false but R is true.
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Q4:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): Sum of all 11 terms of an A.P whose middle most term is 30 is 330.
Reason (R): Sum of first n terms of an A.P is given by l is the middle term
Explanation
Middle most term =
a6 = a + 5d = 30
= 11 × 30 = 330
Also, Sum of first n terms of an A.P is given by is last term(Here we can’t apply this formula as no last term is mentioned in the statement.)
Therefore, A is true but R is false.
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Q5:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): If the nth term of an A.P. is 7 – 4n, then its common differences is – 4.
Reason (R): Common differences of an A.P. is given by d = an + 1 – an
Explanation
We are given, a
n = 7 – 4n
Also, an + 1 = 7 – 4(n + 1) = 7 – 4n – 4 = 3 – 4n
d = a(n + 1)–an
= 3 – 4n – 7 + 4n = – 4
Therefore, Both A and R are true and R is the correct explanation for A.
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Q6:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): If five times the fifth term of an A.P. is equal to eight times its eighth term, then its 13th term is zero.
Reason (R): Common difference of the A.P. a, a + d, a + 2d, …….., is given by d = 2nd term – 1st term.
Explanation
Let the first term and common difference of the A.P. are a and d
5(a + 4d) = 8(a + 7d)
5a + 20d = 8a + 56d
3a + 36d = 0
a + 12d = 0
So, a13 = 0, which is true
Also, the expression d = 2nd term – 1st term
Therefore, Both A and R are true and R is not correct explanation for A.
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Q7:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): There will be 10 rows in the flower bed if there are 23 plants in the first row, 21 in the second row, 19 in the third and so on. Also 5 rose plants are in the last row.
Reason (R): Sum of first n terms of an A.P is given by
Explanation
In case Assertion: According to given statement, Number of rows = 10 Number plants in rows, can be expressed in A.P.
i.e., 23, 21, 19, .........
a = 23
d = 21 – 23 = – 2
an = a + (n – 1)d
a10 = 23 + (10 – 1) (– 2)
a10 = 23 – 18
a10 = 5
∴ a10 = 5 = last row
So, Assertion (A) is correct.
As we are not given any information how many total plants are there so we can’t apply
Therefore, Both A and R are true and R is not correct explanation for A.
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Q8:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: Directions: In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The sum of first n terms of an A.P is given by Sn = 3n2 – 4n. Then its n term is given by the term an = 6n – 7.
Reason (R): nth term of an A.P, whose sum to n terms is Sn, is given by an = Sn – Sn – 1
Explanation
To verify the reason, we are given Sn = 3n
2 – 4n
Then, an – 1 = 3(n – 1)2 – 4(n – 1)
= 3n2 + 3 – 6n – 4n + 4 = 3n2 – 10n + 7
an = Sn – Sn – 1an = 3n2 – 4n – (3n2 – 10n + 7) = 6n – 7
Reason is correct.
Therefore, Both A and R are true and R is the correct explanation for A.
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Q9:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:Assertion : Common difference of the AP -5, -1, 3, 7, .......... is 4.
Reason : Common difference of the AP a,ad +a , d+ 2 , .......... is given by d = 2 nd term-1st term.
Explanation
Common difference, d = - 1-1(- 5) = 4
So, both A and R are correct and R explains A.
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Q10:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:Assertion : an - an - 1 is not independent of n then the given sequence is an AP.
Reason : Common difference d =an - an - 1 is constant or independent of n .
Explanation
We have, common difference of an AP
d =an - an - 1 is independent of n or constant.
So, A is correct but R is incorrect.
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Q11:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:Assertion : Three consecutive terms 2k +1, 3k+3and 5k -1 form an AP than k is equal to 6.
Reason : In an AP a,ad +a , d+ 2 , .........., the sum to n terms of the AP be
Explanation
For 2k +1, 3k + 3 and 5k -1 to form an AP
(3k + 3) - (2k + 1) = (5k -1) - (3k + 3)
k + 2 = 2k - 4
2 + 4 = 2k - k = k
k = 6
So, both A and R are correct but R does not explain A.
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Q12:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:Assertion : The sum of the first n terms of an AP is given by Sn = 3n2 -4 . Then n its nth term an = 6n -7 .
Reason : n th term of an AP, whose sum to n terms is Sn , is given by an =Sn - Sn - 1
Explanation
n th term of an AP be a
n = S
n - S
n - 1
an = 3n2 - 4n- 3(n - 1)2 + 4)(n - 1)
an = 6n - 7
So, both A and R are correct and R explains A.
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Q13:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:Assertion : Common difference of an AP in which a21 -a7 = 84 is 14.
Reason : n th term of AP is given by an = a + (n - 1)d
Explanation
We have, a
n = a + (n - 1)d
a21 -a7 = {a + (21 - 1)d}
- {a + (7 - 1)d} = 84
a + 20d- a - 6d = 84
14d = 84
d = 18/14 = 6
d = 6
So, A is incorrect but R is correct
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Q14:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:Assertion : Arithmetic between 8 and 12 is 10.
Reason : Arithmetic between two numbers ‘a ’ and 'b' is given as
Explanation
Both are correct and Reason is the correct explanation for the Assertion.
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Q15:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself:Direction : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:
Assertion : Let the positive numbers a,b,c be in A.P., then 1/bc, 1/ac, 1/ab are also in A.P.
Reason : If each term of an A.P. is divided by abc, then the resulting sequence is also in A.P.
Explanation
if a,b,c are in AP, then b − a = c − b
a + c = 2b
For 1/bc, 1/ac, 1/ab
= 1/ac - 1/bc = b - a / abc
1/ac - 1/ac = c - a/abc
From above two equations,
1/bc, 1/ac, 1/ab are also in AP.
And their common difference is divided by abc since numbers are also divided by abc.
So, if a,b,c are in AP and you divide the numbers by their product abc they will remain in AP.
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Q16:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:Assertion : Sum of first 10 terms of the arithmetic progression - 0.5 , - 1.0, - 1.5 , .......... is 27.5
Reason : Sum of n terms of an A.P. is given as where a = first term, d = common difference.
Explanation
Both are correct. Reason is the correct reasoning for Assertion.
Assertion,
= 5 [- 1 - 4.5]
= 5 (- 5.5) = 27.5
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Q17:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:Assertion : The sum of the series with the nth term. tn = (9 - 5n) is (465), when no. of terms n = 15 .
Reason : Given series is in A.P. and sum of n terms of an A.P. is
Explanation
Given: a
n = 9 – 5n
Taking n = 1,
a1 = 9 – 5(1) = 9 – 5 = 4
Taking n = 2,
a2 = 9 – 5(2) = 9 – 10 = -1
Taking n = 3,
a3 = 9 – 5(3) = 9 – 15 = -6
Therefore the series is 4, -1, -6, …
So, a = 4, d = a2 – a1 = -1 – 4 = -5
Now, we have to find the sum of the first 15 terms of the AP
⇒ S15 = 15 × (-31)
⇒ S15 = -465
Hence, the sum of 15 terms is -465.
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Q18:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:Assertion : If n th term of an A.P. is 7 -4n , then its common differences is -4.
Reason : Common difference of an A.P. is given by d =an + 1 - an
Explanation
Both are correct. Reason is the correct explanation.
Assertion, an = 7 - 4n
d = an + 1 -an
= 7 - 4 (n + 1) - 7 (-4n)
= 7 - 4n - 4 - 7 + 4n = - 4
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Q19:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:Assertion : If Sn is the sum of the first n terms of an A.P., then its nth term an is given by an =Sn - Sn - 1 .
Reason : The 10th term of the A.P. 5, 8, 11, 14, .......... is 35.
Explanation
Assertion (A) is true but reason (R) is false.
a10 = a + 9d
= 5 + 9(3) = 5 + 27 = 32
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Q20:
Question for Assertion & Reason Type Questions: Arithmetic Progressions
Try yourself: DIRECTION : In the following questions, a statement of assertion (A) is followed by a statement of reason (R). Mark the correct choice as:Assertion : Sum of first hundred even natural numbers divisible by 5 is 500.
Reason : Sum of first n -terms of an A.P. is given by where ℓ = last term.
Explanation
Assertion : Even natural numbers divisible by 5 are 10, 20, 30, 40, ..........
They form an A.P. with,
a = 10 , d = 10
Reason is correct.
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