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Question
Answer the following:
Find 2 × 6 + 4 × 9 + 6 × 12 + ... upto n terms
Solution
2 × 6 + 4 × 9 + 6 × 12 + ... upto n terms
Now, 2, 4, 6, … are in A.P.
∴ rth term = 2 + (r – 1) (2) = 2r
6, 9, 12. … are in A.P.
∴ rth term = 6 + (r – 1) (3) = (3r + 3)
∴ 2 × 6 + 4 × 9 + 6 × 12 + … upto n terms
= `sum_("r" = 1)^"n" 2"r" xx (3"r" + 3)`
= `6sum_("r" = 1)^"n""r"^2 + 6sum_("r" = 1)^"n""r"`
= `6*("n"("n" + 1)(2"n" + 1))/6 + 6("n"("n" + 1))/2`
= 2(n + 1) [2n + 1 + 3]
= 2n(n + 1)(n + 2)
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