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In an A.P., If the First Term is 22, the Common Difference is −4 and the Sum To N Terms is 64, Find N. - CBSE Class 10 - Mathematics

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Question

In an A.P., if the first term is 22, the common difference is −4 and the sum to n terms is 64,  find n.

Solution

In the given problem, we need to find the number of terms of an A.P. Let us take the number of terms as n.

Here, we are given that,

a = 22

d = -4

S_n= 6

So, as we know the formula for the sum of n terms of an A.P. is given by,

`S_n = n/2 [2a + (n - 1)d]`

Where; a = first term for the given A.P.

d = common difference of the given A.P.

= number of terms

So, using the formula we get,

`S_n= n/2 [2(22) + (n - 1)(-4)]`

`64 = n/2[44 - 4n + 4]`

64(2) = n(48 - 4n)

`128 = 48n - 4n^2`

Further rearranging the terms, we get a quadratic equation,

`4n^2 - 48n + 128 = 0`

On taking 4 common we get

`n^2 - 12n + 32 = 0`

Further, on solving the equation for n by splitting the middle term, we get,

`n^2 - 12n + 32 = 0`

`n^2 - 8n  -4n + 32 = 0`

n(n - 8) - 4(n - 8) = 0

(n - 8)(n - 4) = 0

So, we get,

(n - 8) = 0

n = 8

Also

(n - 4) = 0

n = 4

Therefore n = 4 or 8

 

 

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Solution In an A.P., If the First Term is 22, the Common Difference is −4 and the Sum To N Terms is 64, Find N. Concept: Sum of First n Terms of an AP.
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