Question

the sum n terms of an ap is 136 the common difference is 4 and last term is 31 then find the number of terms

Answer 1

Use ap sum and general term formulae.

Answer 2

Answer:

The number of terms are 8.    

Step-by-step explanation:

Given : The sum n terms of an A.P is 136 the common difference is 4 and last term is 31.

To find : The number of terms?

Solution :

Common difference d=4

last term l=31

Sum of n terms is $S_n=136$

The last term formula is

$a_n=a+(n-1)d\\\\31=a+(n-1)4\\\\31=a+4n-4\\\\31+4=a+4n\\\\35=a+4n$

$a=35-4n$ ....(1)

The Sum of n terms formula is

$S_n=\frac{n}{2}(2a+(n-1)d)$

$136=\frac{n}{2}(2a+(n-1)4)$

$136\times 2=n(2a+(n-1)4)$

$372=n(2a+4n-4)$

Substitute the value of equation from (1),

$372=n(2(35-4n)+4n-4)$

$372=n(70-8n+4n-4)$

$372=n(70-4n-4)$

$372=n(66-4n)$

$4n^2-66n+372=0$

$2n^2-33n+136=0$

Solving by quadratic formula,

$n=8,n=\frac{17}{2}$

The value of n can't be in fraction or decimal.

Therefore, The number of terms are 8.