Question

2. Divide 32 into four parts which are the four terms of an AP such that the
product of the first and the fourth terms is to the product of the second
and the third terms as 7:15.
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Answer 1

$\huge\textbf{\underline{As per Question:-}}$

Let the four terms of A.P be

a +3d , a + d , a - d , a - 3d

$\huge\textbf{\underline{Solution:-}}$

• Sum of four term is 32.

$\mathsf{a + 3d + a+d + a-d + a-3d = 32}$

$\mathsf{4a = 32}$

$\mathsf{a = \dfrac{32}{4}}$

$\mathsf{a = 8}$

• Ratio of the product of first term and fourth term to the product of second and third term is 7 : 15.

$\mathsf{\dfrac{(a+3d)(a-3d)}{(a+d)(a-d)} = \dfrac{7}{15}}$

$\mathsf{\dfrac{a^2 - 9d^2}{a^2 -d^2}=\dfrac{7}{15}}$

$\mathsf{15(a^2 -9d^2)= 7(a^2 -d^2)}$

$\mathsf{15a^2 - 135d^2 = 7a^2 - 7d^2 }$

$\mathsf{15a^2 - 7a^2 = -7d^2 +135d^2 }$

$\mathsf{8a^2 = 128d^2 }$

$\mathsf{ 8(8)^2 = 128 d^2}$

$\mathsf{ d^2 = \dfrac{8\times 64}{128}}$

$\mathsf{d^2 = \dfrac{512}{128}}$

$\mathsf{ d^2 = 4}$

$\mathsf{d = \sqrt{4}}$

$\mathsf{\therefore d = 2}$

The divided terms are:-

$\rightarrow$a + 3d = 8 + 3 × 2 = 8 + 6 = 14

$\rightarrow$a - 3d = 8 - 3 × 2 = 8 - 6 = 2

$\rightarrow$a + d = 8 + 2 = 10

$\rightarrow$a - d = 8 - 2 = 6

Answer 2

AnswEr :

• Four Terms are in AP whose sum is 32.
• Product of 1st and 4th terms to product of 2nd and 3rd terms is in ratio 7:15
• Find the Terms.

• Let the Terms in AP be :

(a + 3d) , (a + d) , (a - d) and, (a - 3d)

• According to Question :

⇒ Sum of these Terms = 32

⇒ a + 3d + a + d + a - d + a - 3d = 32

⇒ 4a + 4d - 4d = 32

⇒ 4a = 32

• Dividing Both term by 4

⇒ a = 8

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• According to Question :

⇒ (1st × 4th) / (2nd × 3rd) = 7 / 15

⇒ (a + 3d) × (a - 3d) / (a + d) × (a - d) = 7 / 15

• (a + b) × (a - b) = (a² - b²)

⇒ (a² - 9d²) / (a² - d²) = 7 / 15

• By Cross Multiplication

⇒ 15 × (a² - 9d²) = 7 × (a² - d²)

⇒ 15a² - 135d² = 7a² - 7d²

⇒ 15a² - 7a² = 135d² - 7d²

⇒ 8a² = 128d²

• Using the Value of a = 8

⇒ 8 × (8)² = 128d²

⇒ (8 × 8 × 8) / 128 = d²

⇒ d² = 4

• Squaring Root Both Side

⇒ d = 2

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◑ 1st = (a + 3d) = 8 + 3 × 2 = 8 + 6 = 14

◑ 2nd = (a + d) = 8 + 2 = 10

◑ 3rd = (a - 3d) = 8 - 3 × 2 = 8 - 6 = 2

◑ 4th = (a - d) = 8 - 2 = 6

჻ Required Terms are 14, 10, 2 and 6.