Question

How many terms of the AP 9, 17, 25 must be taken to sum to get a sum of 450?

Answer 1

Answer:

AP = 9, 17, 25 ...

Here the first term ( a ) is 9. Common difference ( d ) is 8

Sum of n terms is 450. To find the value of n = ?

$S_n = \dfrac{n}{2} [ 2a + ( n - 1 ) d\\\\\implies 450 = \dfrac{n}{2} \: [ 2 ( 9 ) + ( n - 1 ) 8 ]\\\\\implies 450 \times 2 = n [ 18 + 8n - 8 ]\\\\\implies 900 = n [ 10 + 8n ]\\\\\implies 900 = 10n + 8n^2 \\\\\implies 8n^2 + 10n - 900 = 0$

Dividing the whole equation by 2 we get,

=> 4n² + 5n - 450 = 0

Solving this further we get,

=> 4n² + 45n - 40n - 450 = 0

=> n ( 4n + 45 ) - 10 ( 4n + 45 ) = 0

=> ( 4n + 45 ) ( n - 10 )

=> n = -45 / 4 or 10

Since number of terms cannot be negative, -45/4 is ignored.

Hence the number of terms in the AP is 10.

Hence 10 terms must be taken to get a sum of 450.

Answer 2

$\underline{\underline{\mathfrak {\large{Solution : }}}}$



$\textsf{Given A.P : 9 , 17 , 25 , ............. }$




$\mathsf{Here,} \\ \\ \mathsf{\implies First \: term (a) \: = \: 9} \\ \\ \mathsf{\implies Common \: difference (d) \: = \: 17 \: - \:9 \:} \\ \\ \mathsf{ \qquad \qquad \qquad \qquad \qquad \qquad \: \: = \: 8 }$




$\mathsf{According \: to \: the \: question, } \\ \\ \mathsf{\implies S_{n} \: = \: 450}$




$\textsf{Using Formula : } \\ \\ \boxed{\mathsf{\implies S_{n} \: = \: \dfrac{n}{2}[2a \: + \: (n \: - \: 1 )d ]}}$



$\mathsf{Now,} \\ \\ \mathsf{\implies 450 \: = \: \dfrac{n}{2}[2 \: \times \: 9 \: + \: ( n \: - \: 1 )8]} \\ \\ \mathsf{\implies 450 \: = \: \dfrac{n}{2}( 18 \: + \: 8n \: - \: 8 )} \\ \\ \mathsf{\implies 450 \:= \: \dfrac{ n}{2}(8n \: + \: \: 10)} \\ \\ \mathsf{ \implies 450 \: = \: \left( \dfrac{n}{ \cancel{2}} \right) \: \times \: \cancel{2}(4n \: + \: 5 )}$



$\mathsf{\implies 450\: = \: n(4n \: + \: 5 ) } \\ \\ \mathsf{\implies 450 \: = \: 4{n}^{2} \: + \: 5n } \\ \\ \mathsf{\implies 0 \: = \: 4{n}^{2} \: + \: 5n \: - 450 } \\ \\ \mathsf{\implies 4{n}^{2} \: + \: 5n \: - \: 450 \: = \: 0 } \\ \\ \mathsf{ \implies 4 {n}^{2} \: + \: (45 \: - \: 40)n \: - \: 450 \: = \: 0}$



$\mathsf{\implies 4{n}^{2} \: + \: 45n \: - \: 40n \: - \: 450 \: = \: 0} \\ \\ \mathsf{\implies n(4n \: + \: 45 ) \: - \: 10(4n \: + \: 45 ) \: = \: 0 } \\ \\ \mathsf{\implies (4n \: + \: 45 ) ( n \: - \: 10) \: = \: 0 }$



$\textsf{By Zero Product Rule : } \\ \\ \mathsf{\implies ( 4n \: + \: 45 ) \: = \: 0 \quad or \: \implies ( n \: - \: 10 ) \: = \: 0}$



$\mathsf{\implies 4n \: = \: -45 \quad or \: \implies n \: = \: 10} \\ \\ \mathsf{\therefore n \: = \: \dfrac{-45}{4} \: [Not \: Possible] \qquad or \qquad \therefore \: \: n \: = \: 10 }$





$\boxed{\textsf{Hence, no. of terms = 10}}$