How many terms of the AP: …

Let the number of terms required to make the sum of 636 be n and common difference be d.

Given Arithmetic Progression : 9 , 17 , 25 ....

First term = a = 9

Second term = a + d = 17

Common difference = d = a + d - a = 17 - 9 = 8

From the indentities of arithmetic progressions, we know : -
{tex}S_{n}=\dfrac{n}{2}\{2a+(n-1)d\}, where S_{n} {/tex}is the sum of first term to nth term of the AP, a is the first term, d is the common difference and n is the number of terms of AP.

In the given Question, sum of APs is 636.

Therefore,
 ⇒ {tex}636 = \dfrac{n}{2} \{2(9) + (n - 1)8 \} \\ \\ \\ = > 1272 = n(18 + 8n - 8) \\ \\ = > 1272 = 10n + 8n {}^{2} \\ \\ = > 636 = 5n + 4n {}^{2}{/tex}

 ⇒ 4n² + 5n - 636 = 0

 ⇒ 4n² + ( 53 - 48 )n - 636 = 0

 ⇒ 4n² + 53n - 48n - 636 = 0

 ⇒ 4n² - 48n + 53n - 636 = 0

 ⇒ 4n( n - 12 ) + 53( n - 12 ) = 0

 ⇒ ( n - 12 )( 4n + 53 ) = 0

By Zero Product Rule,

 ⇒  n - 12 = 0

 ⇒ n = 12

Hence,
Number of terms of the AP [ 9 , 17 , 25 ] which are required to make the sum of 636 is 12.