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प्रश्न
Answer the following:
Find 2 × 5 × 8 + 4 × 7 × 10 + 6 × 9 × 12 + ... upto n terms
उत्तर
2, 4, 6, … are in A.P.
∴ rth term = 2 + (r – 1)2 = 2r
5, 7, 9, … are in A.P
∴ rth term = 5 + (r – 1) (2) = (2r + 3)
8, 10, 12, … are in A.P.
∴ rth term = 8 + (r – 1) (2) = (2r + 6)
2 × 5 × 8 + 4 × 7 × 10 + 6 × 9 × 12 + … to n terms
= `sum_("r" = 1)^"n" 2"r"(2"r" + 3) (2"r" + 6)`
= `sum_("r" = 1)^"n" (8"r"^3 + 36"r"^2 + 36"r")`
= `8sum_("r" = 1)^"n" "r"^3 + 36sum_("r" = 1)^"n" "r"^2 + 36sum_("r" = 1)^"n" "r"`
= `8*("n"^2("n" + 1)^2)/4 + 36*("n"("n" + 1)(2"n" + 1))/6 + 36*("n"("n" + 1))/2`
= 2n (n + 1) [n(n + 1) + 3(2n + 1) + 9]
= 2n (n + 1)(n2 + 7n + 12)
= 2n (n + 1) (n + 3) (n + 4)
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