Question

5.
Four numbers are in A.S. The sum of the fin
& fourth is 14 and the product of the secimi
and third is 45. Find the numbers.​

Answer 1

✪ᴀɴsᴡᴇʀ✪

• The four numbers are $1, 5, 9, 13\:or\:13, 9, 5, 1$

★ᴇxᴘʟᴀɪɴᴀᴛɪᴏɴ★

Let $a - 3d, a-d, a+d, a+3d$ be the four numbers in AP.(2d is the common difference).

Then,

Sum of the first and fourth is 14, so

$\:\:\:\:\:\:\: (a - 3d) + (a+3d) = 14$

$\implies 2a = 14$

$\implies a = 7$

Also,

Product of the second and third is 45,so

$\:\:\:\:\:\:\: (a - d)(a+d) = 45$

$\implies a^2 - d^2 = 45$

$\implies 7^2 - d^2 = 45$

$\implies d^2 = 49 - 45$

$\implies d^2 = 4$

$\implies d = ± 2$

Hence, the four numbers are

when d = 2

$\to 7-3.2, 7-2, 7+2, 7+3.2$

$\to 7-6, 7-2, 7+2, 7+6$

$\to 1 , 5, 9, 13$

when d = - 2

$\to 7 + 3.2, 7+2, 7-2, 7-3.2$

$\to 7 + 6, 7+2, 7-2, 7-6$

$\to 13, 9, 5, 1$