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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

ConceptSum of First n Terms of an AP

#### 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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