Question

if the sum of first four terms of an ap is 32. the ratio of product the first term and fourth term is 2 product of middle term is 7 ratio 15 .find the numbers

Answer 1

Answer:

→ 2, 6, 10, 14 .

Step-by-step explanation:

Note :- This question is come in CBSE class 10th board 2018 .

Solution:-

Let the four consecutive numbers in AP be (a - 3d), (a - d), (a + d) and (a + 3d)

So, according to the question.

⇒ a-3d + a - d + a + d + a + 3d = 32

⇒ 4a = 32

⇒ a = 32/4

∵ a = 8 ......(1)

Now, (a - 3d)(a + 3d)/(a - d)(a + d) = 7/15

⇒ 15(a² - 9d²) = 7(a² - d²)

⇒ 15a² - 135d² = 7a² - 7d²

⇒ 15a² - 7a² = 135d² - 7d² 

⇒ 8a² = 128d²

Putting the value of a = 8 in above we get.

⇒ 8(8)² = 128d²

⇒ 128d² = 512

⇒ d² = 512/128

⇒ d² = 4

∴ d = 2

So, the four consecutive numbers are

⇒ a - 3d = 8 - (3×2 )=  8 - 6 = 2.

⇒ a - d = 8 - 2 = 6.

⇒ a + d = 8 + 2 = 10.

⇒ a + 3d = 8 + (3×2) = 8 + 6 = 14.

Four consecutive numbers are 2, 6, 10 and 14

Hence, it is solved .

THANKS .

Answer 2

Que :- If the sum of first four terms of an ap is 32. the ratio of product the first term and fourth term is 2 product of middle term is 7 ratio 15 .find the numbers?

Solution:-

Let the four consecutive numbers be (a-3d), (a+3d) , (a-d) and (a + d).

Now, Sum of the four consecutive term is 32.

=> a - 3d + a + 3d + a - d + a + d = 32
=> 4a = 32
=> a = 32/4
=> a = 8 ________________(1)

A.T.Q.

$= > \frac{(a - 3d)(a + 3d)}{(a - d)(a + d)} = \frac{7}{15} \\ \\ = > \frac{ {a}^{2} - {9d}^{2} }{ {a}^{2} - {d}^{2} } = \frac{7}{15} \\ \\ = > 15( {a}^{2} - {9d}^{2} ) = 7( {a}^{2} - {d}^{2} ) \\ \\ = > 15 {a}^{2} - 135 {d}^{2} = 7 {a}^{2} - 7 {d}^{2} \\ \\ = > 15 {a}^{2} - 7 {a}^{2} = 135 {d}^{2} - 7 {d}^{2} \\ \\ = > 8 {a}^{2} = 128 {d}^{2} \\ \\ putting \: a = 8 \: here.we \: get \\ \\ = > 8 \times {8}^{2} = 128 {d}^{2} \\ \\ = > 512 = 128 {d}^{2} \\ \\ = > {d}^{2} = \frac{512}{128} \\ \\ = > {d}^{2} = 4 \\ \\ = > d = 2.$

Hence,

The Four Consecutive Numbers are :-

a - 3d = 8 - 6 = 2.

a + 3d = 8 + 6 = 14.

a - d = 8 - 2 = 6

a + d = 8 + 2 = 10.