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Question
If the sum of n terms of an A.P. is Sn = 3n2 + 5n. Write its common difference.
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Solution
Here, we are given,
`S_n = 3n^2 + 5n`
Let us take the first term as a and the common difference as d.
Now, as we know,
`a_n = S_n - S_(n-1)`
So, we get,
`a_n = (3n^^^^2 + 5n) - [3(n-1)^2 + 5 (n-1)]`
`=3n^2 + 5n - [3(n^2 + 1 - 2n) + 5n - 5] [\text{ Using} (a - b)^2= a^2 - ab]`
`=3n^2 + 5n - (3n^2 + 3 - 6n + 5n - 5)`
`=3n^2 + 5n - 3n^2 - 3 + 6n - 5n + 5`
= 6n + 2 ..................(1)
Also,
`a_n = a + (n-1)d`
= a + nd - d
= nd + ( a- d) ...............(2)
On comparing the terms containing n in (1) and (2), we get,
dn = 6n
d = 6
Therefore, the common difference is d = 6 .
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