Name: General Term of an AP: Example
Uploaded: 2019-07-19
Duration: 4 min 25 s
Description: Do you know how to find the first five terms of a sequence in A.P with the help of any two given terms of the sequence? Watch this video to learn how.

Question

The sum of four consecutive numbers in an

A P

is

32

and the ratio of the product of the first and the last term to the product of two middle terms is

7 : 15

. Find the numbers.

Accepted Answer

Let the four consecutive number be $a, a + d, a + 2 d, a + 3 d$

Given, that sum of four consecutive numbers in $A . P$ is $32$ .

Therefore,

$a + a + d + a + 2 d + a + 3 d = 32 4 a + 6 d = 32 2 a + 3 d = 16 a = \frac{16 - 3 d}{2} . . . . . . . . . (1)$

The ratio of the product of the first and the last term to the product of two middle term is $7 : 15$ .

Therefore,

$\frac{a (a + 3 d)}{(a + d) (a + 2 d)} = \frac{7}{15} \Rightarrow 15 a (a + 3 d) = 7 (a + d) (a + 2 d) \Rightarrow 15 a^{2} + 45 a d = 7 a^{2} + 21 a d + 14 d^{2} \Rightarrow 8 a^{2} + 24 a d - 14 d^{2} = 0 \Rightarrow 4 a^{2} + 12 a d - 7 d^{2} = 0 \Rightarrow 4 a^{2} + 14 a d - 2 a d - 7 d^{2} = 0 \Rightarrow 2 a (2 a + 7 d) - d (2 a + 7 d) = 0 \Rightarrow (2 a + 7 d) (2 a - d) = 0 ∴ a = \frac{- 7 d}{2} & a = \frac{d}{2} . . . . . . . . . . . (2)$

Substitute the value of $a$ in equation $(2)$

So,

$a = \frac{- 7 d}{2} \Rightarrow \frac{16 - 3 d}{2} = \frac{- 7 d}{2} \Rightarrow - 4 d = 16 \Rightarrow d = - 4$

and

$a = \frac{d}{2} \Rightarrow \frac{16 - 3 d}{2} = \frac{d}{2} \Rightarrow 4 d = 16 \Rightarrow d = 4$

When $d = 4$

Then, $a = \frac{d}{2} = \frac{4}{2} = 2$

Therefore, four consecutive number in an $A P$ are

$2, (2 + 4), (2 + 2 \times 4), (2 + 3 \times 4) = 2, 6, 10, 14$

If we take $d = - 4$ then also we will get same numbers.

The sum of four consecutive numbers in an $A P$ is $32$ and the ratio of the product of the first and the last term to the product of two middle terms is $7 : 15$ . Find the numbers.

Important Questions on Arithmetic Progressions

Learn and Prepare for any exam you want !

The sum of four consecutive numbers in an AP is 32 and the ratio of the product of the first and the last term to the product of two middle terms is 7 : 15. Find the numbers.

Important Questions on Arithmetic Progressions

Learn and Prepare for any exam you want !

The sum of four consecutive numbers in an $A P$ is $32$ and the ratio of the product of the first and the last term to the product of two middle terms is $7 : 15$ . Find the numbers.