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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.
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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.
n = 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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