The sum of the first three terms of an Arithmetic Progression (A.P.) is 42 and the product of the first and third term is 52 . Find the first term and the common difference.

Let the three terms of the A.P. be a - d , a , a + d , where d is the common difference. The sum of first three terms of the A.P. is 42 Therefore, a - d + a + a + d = 42 3 a = 42 a = 42 3 = 14 Also given that, product of first and third term of the A.P. is 52 ( a - d ) ( a + d ) = 52 a 2 - d 2 = 52 14 2 - d 2 = 52 ( ∵ a = 14 ) 196 - d 2 = 52 d 2 = 196 - 52 d 2 = 144 d = 12 or d = - 12 Hence, the first term of the A.P. is 2 and the common difference is 12 or - 12 .

The 4 th , 6 th and the last term of a geometric progression are 10 , 40 and 640 respectively. If the common ratio is positive, find the first term, common ratio and the number of terms of the series.

Let a be the first term and r be the common ratio of the GP. We know that n th term = a n = a r n - 1 So, a r 3 = 10 ...... i and a r 5 = 40 ..... i i If we divide i i by i , we get a r 5 a r 3 = 40 10 ⇒ r 2 = 4 ⇒ r = ± 2 Since given that the common ratio is positive, so r = 2 Put r = 2 in equation i , we get a × 2 3 = 10 ⇒ a = 10 8 = 5 4 Now, given that last term is 640 . So, a n = 640 ⇒ a r n - 1 = 640 ⇒ 5 4 × 2 n - 1 = 640 ⇒ 2 n - 1 = 640 × 4 5 ⇒ 2 n - 1 = 512 ⇒ 2 n - 1 = 2 9 ⇒ n - 1 = 9 [Since a m = a n ⇒ m = n ] ⇒ n = 10 Hence, common ratio 2 , first term is 5 4 and number of terms is 10 .

E M B I B E

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 9 - Arithmetic and Geometric Progression>ICSE-2020>Q 11

HARD

10th ICSE

IMPORTANT

Earn 100

If the $6^{t h}$ term of an A.P. is equal to four times its first term and the sum of first six terms is $75$ , find the first term and the common difference.

Important Questions on Arithmetic and Geometric Progression

EASY

10th ICSE

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 9 - Arithmetic and Geometric Progression>ICSE-2018>Q 1

If

(k - 3), (2 k + 1)

and

(4 k + 3)

are three consecutive terms of an A.P., find the value of

k

?

HARD

10th ICSE

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 9 - Arithmetic and Geometric Progression>ICSE-2018>Q 2

The $4^{th}$ term of a G.P. is $16$ and the $7^{th}$ term is $128$ . Find the first term and common ratio of the series.

HARD

10th ICSE

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 9 - Arithmetic and Geometric Progression>ICSE-2018>Q 3

The $4^{th}$ term of an arithmetic progression is $22$ and $15^{th}$ term is $66$ . Find the first term and the common difference. Hence, find the sum of the series to $8$ terms.

MEDIUM

10th ICSE

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 9 - Arithmetic and Geometric Progression>ICSE-2019>Q 4

In an Arithmetic progression (A.P.), the fourth and sixth terms are $8$ and $14$ respectively. Find the first term.

EASY

10th ICSE

IMPORTANT

EMBIBE CHAPTER WISE PREVIOUS YEAR PAPERS FOR MATHEMATICS>Chapter 9 - Arithmetic and Geometric Progression>ICSE-2019>Q 5

In an Arithmetic Progression (A.P.) the fourth and sixth terms are $8$ and $14$ respectively. Find the common difference.

EASY

10th ICSE