Question

in AP the sum of 4th and 10th term is 10 and multiplication of these terms is 24 find the sum of first 25 terms​

Answer 1

The general term of an AP,

$a_{n}$ = a + (n-1)d

where a is the first term and d is the common difference

a₄ = a + 3d

a₁₀ = a + 9d

a₄ + a₁₀ = 2a + 12d = 10

a + 6d = 5

a₄×a₁₀ = (a+3d)(a+9d)

          = a²+12ad+27d²

          = a²+12ad+36d²-9d²

          = (a+6d)² - 9d²

But a₄×a₁₀ = 24

Hence (a+6d)² - 9d² = 24

But we have  a + 6d = 5

Therefore,

(a+6d)² - 9d² = 24

25-9d² = 24

9d² = 1

d = ± $\frac{1}{3}$

When d = $\frac{1}{3}$, a = 3, $a_{25}$ = 11

When d = -$\frac{1}{3}$, a = 7, $a_{25}$ = -1

$S_{n}$ = $\frac{n(a + a_{n})}{2}$

When d =  $\frac{1}{3}$, $S_{25}$ = 175

When d =  -$\frac{1}{3}$, $S_{25}$ = 75

HOPE IT HELPS !!