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Question
Find the sum 22 + 42 + 62 + 82 + ... upto n terms
Solution
The numbers 2, 4, 6, 8, ... are in A.P. with a = 2, d = 2.
∴ the rth term= a + (r – 1)d
= 2 + (r – 1)2 = 2r
∴ rth term of given series = tr = (2r)2 = 4r2
∴ 22 + 42 + 62 + 82 + ... upto n terms.
= `sum_("r" = 1)^"n" "t"_"r"`
= `sum_("r" = 1)^"n"(4"r"^2)`
= `4sum_("r" = 1)^"n" "r"^2`
= `4.("n"("n" + 1)(2"n" + 1))/6`
= `(2"n"("n" + 1)(2"n" + 1))/3`
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