Test: Arithmetic And Geometric Progressions - 1 - CA Foundation MCQ

Let a be the first term and d be the common difference.
Given series is in AP.

a = 1
d = 3 - 1 = 2

We know that nth term of an Ap an = a + (n - 1) * d

= 1 + (n - 1) * 2
= 1 + 2n - 2
= 2n - 1.

Correct answer is option C) 50.

Finding the value of 'a':
- Given: The arithmetic mean between 'a' and 10 is 30.
- Formula for arithmetic mean: (a + b) / 2
- Plug in the given values: (a + 10) / 2 = 30

Calculating 'a':
- Multiply both sides by 2: a + 10 = 60
- Subtract 10 from both sides: a = 50
Answer:
- The value of 'a' is 50 (Option C).

Correct answer is option A) -1, 0, 3

To find the first three terms of the sequence, we need to substitute n = 1, n = 2, and n = 3 into the nth term formula, t_n = n^2 - 2n.
First term (n = 1):
- t1 = (1)^2 - 2(1)
- t1 = 1 - 2
- t1 = -1

Second term (n = 2):
- t2 = (2)^2 - 2(2)
- t2 = 4 - 4
- t2 = 0

Third term (n = 3):
- t3 = (3)^2 - 2(3)
- t3 = 9 - 6
- t3 = 3

So, the first three terms of the sequence are:
- -1, 0, 3
Therefore, the correct answer is option a (-1, 0, 3).

Correct answer is option B) 20^th

Finding the term position:

• Given an arithmetic progression: –1, –3, –5, ...
• Common difference (d) between terms is: –3 - (-1) = -2
• First term (a) is: –1
• Term we are looking for is: -39
• Formula for the nth term of an arithmetic progression: a_n = a + (n-1)d
• Substitute the values: -39 = -1 + (n-1)(-2)
• Solve for n: -39 + 1 = (n-1)(-2) => -38 = -2(n-1) => 19 = n-1 => n = 20

Answer: The 20th term of the progression is -39.

Correct answer is option C) 15/2

To find the value of x such that 8x + 4, 6x - 2, and 2x + 7 form an Arithmetic Progression (AP), we can follow these steps:
Step 1: Identify the terms in the AP
- First term (a1) = 8x + 4
- Second term (a2) = 6x - 2
- Third term (a3) = 2x + 7

Step 2: Use the property of AP
In an arithmetic progression, the difference between consecutive terms is constant. Therefore, the difference between the second term and the first term should be equal to the difference between the third term and the second term.
(a2 - a1) = (a3 - a2)

Step 3: Substitute the terms and solve for x
- (6x - 2) - (8x + 4) = (2x + 7) - (6x - 2)
- (-2x - 6) = (9 - 4x)
- (-2x + 4x) = (9 + 6)
- 2x = 1
- x = 15/2
So, the value of x is 15/2, which is option (c) 15/2.

We know that, the sum of infinite terms of GP is

It is given that the sum of n terms is

Correct answer is option A) 58

To find the 20th term of the given arithmetic progression, we can use the formula:

nth term (Tn) = a + (n - 1) * d
where:
- a is the first term
- n is the position of the term in the sequence
- d is the common difference between consecutive terms

For the given progression:
- a = 1 (first term)
- n = 20 (we want to find the 20th term)
- d = 4 - 1 = 3 (common difference)

Applying the formula:
T20 = 1 + (20 - 1) * 3
- T20 = 1 + 19 * 3
- T20 = 1 + 57
- T20 = 58

So, the 20th term of the progression is 58.

• The series given is an arithmetic progression (AP) where the first term (a) is 5 and the common difference (d) is 2.
• The formula for the nth term of an AP is: nth term = a + (n - 1) * d.
• For 21 terms, substitute a = 5, d = 2, and n = 21: nth term = 5 + (21 - 1) * 2.
• This simplifies to: 5 + 20 * 2 = 5 + 40 = 45.
• Thus, the last term of the series is 45.

Let the terms be – 6, a, b 14

a = – 6

T4 = a + 3d = 14

– 6 + 3d = 14

3d = 20

d = 20/3

a = – 6 + 20/3 = 2/3

b = 2/3 + 20/3 = 22/3

let a - d , a , a + d are three terms of an AP

according to the problem given,

sum of the terms = 15

a - d + a + a + d = 15

3a = 15

a = 15 / 3 

a = 5

product = 80

( a - d ) a ( a + d ) = 80

( a² - d² ) a = 80

( 5² - d² ) 5 = 80

25 - d² = 80 /5

25 - d² = 16

- d² = - 9

d² = 3²

d = ± 3

Therefore,

a = 5 , d = ±3

required 3 terms are

a - d = 5 - 3 = 2

a = 5

a+ d = 5 + 3 = 8

( 2 , 5 , 8 ) or ( 8 , 5 , 2 )

The sum of n terms of an A.P. = 3n²+5n

then sum of n-1 terms= 3(n-1)²+5(n-1)

So nth term of A.P will be Tn=3n²+5n- 3(n-1)²-5(n-1)= 3n²+5n-3n²-3+6n-5n+5=6n+2

in this way by putinng values of n= 1,2,3,4,5,6,7,_______n-1, n

we will get the AP as 8,14,20,26,32,38,44,____

a=-2 n=6

given : t6= 23

23= a+5d

23= -2+5d

d= 25/5

d = 5

t2= a+d= -2+5=3

t3= 8

t4= 13

t5= 18