Advertisements
Advertisements
प्रश्न
Find the sum of the series (33 – 23) + (53 – 43) + (73 – 63) + ... to 10 terms
उत्तर
Given series
⇒ (33 – 23) + (53 – 43) + (73 – 63) + ...
= (33 + 53 + 73 + …) – (23 + 43 + 63 + …)
⇒ [33 + 53 + 73 + … (2n + 1)3] – [23 + 43 + 63 + … (2n)3]
∴ Tn = (2n + 1)3 – (2n)3
= (2n + 1 – 2n) [2n + 1)2 + (2n + 1)(2n) + (2n)2] ....[∵ a3 – b3 = (a – b)(a2 + ab + b2)]
= 1 · [4n2 + 1 + 4n + 4n2 + 2n + 4n2]
= 12n2 + 6n + 1
S10 = 4(10)3 + 9(10)2 + 6(10)
= 4 × 1000 + 900 + 60
= 4000 + 960
= 4960
APPEARS IN
संबंधित प्रश्न
Find the sum to n terms of the series 1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + …
Find the sum to n terms of the series 1 × 2 × 3 + 2 × 3 × 4 + 3 × 4 × 5 + …
Find the sum to n terms of the series 3 × 12 + 5 × 22 + 7 × 32 + …
Find the sum to n terms of the series 52 + 62 + 72 + ... + 202
Find the sum to n terms of the series 3 × 8 + 6 × 11 + 9 × 14 +…
Find the sum to n terms of the series 12 + (12 + 22) + (12 + 22 + 32) + …
Find the sum to n terms of the series whose nth terms is given by n2 + 2n
13 + 33 + 53 + 73 + ...
1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + ...
Find the sum of the series whose nth term is:
n3 − 3n
Find the sum of the series whose nth term is:
(2n − 1)2
Find the 20th term and the sum of 20 terms of the series 2 × 4 + 4 × 6 + 6 × 8 + ...
Write the sum of the series 2 + 4 + 6 + 8 + ... + 2n.
Write the sum of the series 12 − 22 + 32 − 42 + 52 − 62 + ... + (2n − 1)2 − (2n)2.
3 + 7 + 14 + 24 + 37 + ...
\[\frac{1}{1 . 6} + \frac{1}{6 . 11} + \frac{1}{11 . 14} + \frac{1}{14 . 19} + . . . + \frac{1}{(5n - 4) (5n + 1)}\]
The value of \[\sum^n_{r = 1} \left\{ (2r - 1) a + \frac{1}{b^r} \right\}\] is equal to
If Sn = \[\sum^n_{r = 1} \frac{1 + 2 + 2^2 + . . . \text { Sum to r terms }}{2^r}\], then Sn is equal to
Write the 50th term of the series 2 + 3 + 6 + 11 + 18 + ...
If \[1 + \frac{1 + 2}{2} + \frac{1 + 2 + 3}{3} + . . . .\] to n terms is S, then S is equal to
Write the sum to n terms of a series whose rth term is r + 2r.
If \[\sum^n_{r = 1} r = 55, \text{ find } \sum^n_{r = 1} r^3\] .
2 + 5 + 10 + 17 + 26 + ...
Find the sum of the series (33 – 23) + (53 – 43) + (73 – 63) + … to n terms
Let Sn denote the sum of the cubes of the first n natural numbers and sn denote the sum of the first n natural numbers. Then `sum_(r = 1)^n S_r/s_r` equals ______.
If |x| < 1, |y| < 1 and x ≠ y, then the sum to infinity of the following series:
(x + y) + (x2 + xy + y2) + (x3 + x2y + xy2 + y3) + .... is ______.
A GP consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places, the common ratio will be equal to ______.
The sum `sum_(k = 1)^20k 1/2^k` is equal to ______.
