Q Find the n terms 6 + 9+16+27 + - Maths - Sequences and Series

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Q Find the n terms 6 + 9+16+27 +

Brijendra Pal · Accepted Answer

Hi, S = 6+9+16+27+
+tnS=       6+9+16+27+
+tn-1+tnSubtracting the corresponding terms 0 = 6+3+7+11+
+tn-tn-1-tntn=6+3+7+11+
+upto n-1 termsnow term under brackets is a AP with a = 3 and d = 4 and n-1 terms tn=6+n-122×3+n-1-1×4  since in AP , Sum = number of terms22×first term+number of terms-1×common difference=6+n-124n-2= 6+n-12n-1= 6+2n2-3n+1=2n2-3n+7so Sn=∑n=1ntn=2∑n=1nn2-3∑n=1nn+∑n=1n7we know that ∑n=1nn2= n6n+12n+1  and ∑n=1n1 = n 
and ∑n=1nn= nn+12 putting we get 
= 2× n6n+12n+1-3×nn+12+7n
= nn+12n+13-3nn+12+7n =2nn+12n+1-9nn+1+42n6
= 2n2+2n2n+1-9n2-9n+42n6 = 4n3+6n2+2n-9n2-9n+42n6
=4n3-3n2+35n6

Q. Find the n terms 6 + 9+16+27 +......................