Question

Find the number of terms of the ap 9 17 25 whose sum is 636

Answer 1

Answer:

17-6=11

difference is11

a is 9

then,

use n=a (n-1)+d

Answer 2

Answer:

$\large{\underline{\boxed{\sf n = 12}}}$

Step-by-step explanation:

Given that;

The terms of A.P are ;

9, 17, 25....here,

• a = 9
• d = (17 - 9) = 8
• Sₙ = 636

Now, we know that ;

$\sf S_n = \frac{n}{2} [2a + (n - 1)d] \\ \\ \implies \sf 636 = \frac{n}{2} [2 \times 9 + (n - 1) \times 8] \\ \\ \implies \sf 636 \times 2 = n(18 + 8n - 8) \\ \\ \implies \sf 1272 = n(10 + 8n) \\ \\ \implies \sf 1272 = 10n + 8 {n}^{2} \\ \\ \implies \sf 8n {}^{2} + 10n - 1272 = 0 \\ \\ \implies \sf 2(4n {}^{2} + 5n - 636) = 0 \\ \\ \implies \sf 4n {}^{2} + 5n - 636 = 0$

Here, we got a quadratic equation, on comparing the given equation with ax² + bx + c= 0,

• a = 4
• b = 5
• c = -636

Discriminant = b² - 4ac

⇒ D = (5)² - 4 * 4 * (-636)

⇒ D = 25 + 10176

⇒ D = 10201

Hence, roots of the equation is given by;

$\tt n = \frac{ - b \pm \sqrt{D} }{2a} \\ \\ \tt n = \frac{ - 5 \pm \sqrt{10201} }{2 \times 4} \\ \\ \tt n = \frac{ - 5 \pm 101}{8}$

Therefore,

$\tt n = \frac{ - 5 + 101}{8} \\ \\ \tt n = \frac{96}{8} \\ \\ \boxed{ \bf \therefore \: n = 12}$

Also,

$\tt n = \frac{ - 5 - 101}{8} \\ \\ \tt n = \frac{ - 106}{8} \\ \\ \boxed{ \bf \therefore \: n = - \frac{53}{4}}$

Since, n cannot be negative.

∴ n = 12

_______________________

Hence, the required number of terms is 12.