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Sum of N Terms of AP And Arithmetic Progression - The sum of n terms of AP is the sum(addition) of first n terms of the arithmetic sequence. It is equal to n divided by 2 times the sum of twice the first term – 'a' and the product of the difference between second and first term-'d' also know as common difference, and (n-1), where n is numbers of terms to be added.
Let a be the first term and d the common difference of the given A.P. Then,
S
2n
=3S
n
sum of n terms−
2
n
[2a+(n−1)d]
⇒
2
2n
[2a+(2n−1)d]=
2
3n
[2a+(n−1)d]
⇒2[2a+(2n−1)d]=3[2a+(n−1)d]
⇒6a−4a−(3n−3−4n+2)d=0
⇒2a−(n+1)d=0
⇒2a=(n+1)d
Now,
S
n
S
3n
=
[(n+1)d+(n−1)d]
3[(n+1)d+(3n−1)d]
=
[nd+nd]
3[nd+3nd]
S
n
S
3n
=
2nd
12nd
=6