Advertisements
Advertisements
प्रश्न
Find the sum 1 × 3 × 5 + 3 × 5 × 7 + 5 × 7 × 9 + ... + (2n – 1) (2n + 1) (2n + 3)
उत्तर
1 × 3 × 5 + 3 × 5 × 7 + 5 × 7 × 9 + ... + (2n – 1) (2n + 1) (2n + 3)
= `sum_("r" = 1)^"n" (2"r" - 1)(2"r" + 1)(2"r"+ 3)`
= `sum_("r" = 1)^"n" (4"r"^2 - 1)(2"r" + 3)`
= `sum_("r" = 1)^"n"(8"r"^3 + 12"r"^2 - 2"r" - 3)`
= `8sum_("r" = 1)^"n" "r"^3 + 12 sum_("r" = 1)^"n" "r"^2 - 2 sum_("r" = 1)^"n" "r" - 3 sum_("r" = 1)^"n" 1`
`=8.("n"^2("n" + 1)^2)/4 + 12*("n"("n" + 1)(2"n" + 1))/6 - 2.("n"("n" + 1))/2 - 3"n"`
= 2n2(n + 1)2 + 2n(n + 1)(2n + 1) – n(n + 1) – 3n
= n[2n(n2 + 2n + 1) + 2(2n2 + 3n + 1) – (n + 1) – 3]
= n(2n3 + 4n2 + 2n + 4n2 + 6n + 2 – n – 1 – 3)
= n(2n3 + 8n2 + 7n – 2)
APPEARS IN
संबंधित प्रश्न
Find the sum to n terms 3 + 33 + 333 + 3333 + …
Find the sum to n terms 8 + 88 + 888 + 8888 + ...
Find the sum to n terms 0.7 + 0.77 + 0.777 + ...
Find Sn of the following arithmetico - geometric sequence:
2, 4x, 6x2, 8x3, 10x4, …
Find Sn of the following arithmetico - geometric sequence:
1, 4x, 7x2, 10x3, 13x4, …
Find Sn of the following arithmetico - geometric sequence:
1, 2 × 3, 3 × 9, 4 × 27, 5 × 81, …
Find Sn of the following arithmetico - geometric sequence:
3, 12, 36, 96, 240, …
Find the sum to infinity of the following arithmetico - geometric sequence:
`1, 2/4, 3/16, 4/64, ...`
Find the sum to infinity of the following arithmetico - geometric sequence:
`3, 6/5, 9/25, 12/125, 15/625, ...`
Find the sum to infinity of the following arithmetico - geometric sequence:
`1, -4/3, 7/9, -10/27 ...`
Find the sum `sum_("r" = 1)^"n" ("r" + 1)(2"r" - 1)`
Find `sum_("r" = 1)^"n"(3"r"^2 - 2"r" + 1)`
Find `sum_("r" = 1)^"n" [(1^3 + 2^3 + .... + "r"^3)/("r"("r" + 1))]`
Find the sum 5 × 7 + 9 × 11 + 13 × 15 + ... upto n terms
Find the sum 22 + 42 + 62 + 82 + ... upto n terms
If `(1 xx 2 + 2 xx 3 + 3 xx 4 + 4 xx 5 + ... "upto n terms")/(1 + 2 + 3 + 4 + ... "upto n terms") = 100/3,` find n
If S1, S2 and S3 are the sums of first n natural numbers, their squares and their cubes respectively then show that - 9S22 = S3 (1 + 8 S1)
Answer the following:
Find 2 + 22 + 222 + 2222 + ... upto n terms
Answer the following:
Find `sum_("r" = 1)^"n" ((1^2 + 2^2 + 3^2 + ... + "r"^2)/(2"r" + 1))`
Answer the following:
Find 2 × 6 + 4 × 9 + 6 × 12 + ... upto n terms
Answer the following:
Find 2 × 5 × 8 + 4 × 7 × 10 + 6 × 9 × 12 + ... upto n terms
Answer the following:
Find `1^2/1 + (1^2 + 2^2)/2 + (1^2 + 2^2 + 3^2)/3 + ...` upto n terms
Answer the following:
If `(1 + 2 + 3 + 4 + 5 + ... "upto n terms")/(1 xx 2 + 2 xx3 + 3 xx 4 + 4 xx5 + ... "upto n terms") = 3/22` Find the value of n
Answer the following:
If `(1 xx 3 + 2 xx 5 + 3 xx 7 + ... "upto n terms")/(1^3 + 2^3 + 3^3 + ... "upto n terms") = 5/9`, find the value of n
Answer the following:
If p, q, r are in G.P. and `"p"^(1/x) = "q"^(1/y) = "r"^(1/z)`, verify whether x, y, z are in A.P. or G.P. or neither.
The sum of n terms of the series 22 + 42 + 62 + ........ is ______.
