Advertisements
Advertisements
प्रश्न
Write the sum of the series 12 − 22 + 32 − 42 + 52 − 62 + ... + (2n − 1)2 − (2n)2.
उत्तर
The given series can be rewritten as:
\[3\left( - 1 \right) + 7\left( - 1 \right) + 11\left( - 1 \right) + . . . + \left( 4n - 1 \right) \left( - 1 \right)\]
\[= - \left[ 3 + 7 + 11 + . . . + \left( 4n - 1 \right) \right]\]
\[ = - \left[ \frac{n}{2}\left\{ 3 \times 2 + \left( n - 1 \right)4 \right\} \right]\]
\[ = - \left[ \frac{n}{2}\left( 4n + 2 \right) \right]\]
\[ = - n\left( 2n + 1 \right)\]
APPEARS IN
संबंधित प्रश्न
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 `1/(1xx2) + 1/(2xx3)+1/(3xx4)+ ...`
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
Find the sum to n terms of the series whose nth terms is given by (2n – 1)2
22 + 42 + 62 + 82 + ...
1.2.5 + 2.3.6 + 3.4.7 + ...
1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + ...
3 × 12 + 5 ×22 + 7 × 32 + ...
Write the sum of the series 2 + 4 + 6 + 8 + ... + 2n.
1 + 3 + 7 + 13 + 21 + ...
1 + 3 + 6 + 10 + 15 + ...
1 + 4 + 13 + 40 + 121 + ...
2 + 4 + 7 + 11 + 16 + ...
\[\frac{1}{1 . 4} + \frac{1}{4 . 7} + \frac{1}{7 . 10} + . . .\]
If ∑ n = 210, then ∑ n2 =
Write the sum of 20 terms of the series \[1 + \frac{1}{2}(1 + 2) + \frac{1}{3}(1 + 2 + 3) + . . . .\]
The sum to n terms of the series \[\frac{1}{\sqrt{1} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{5}} + \frac{1}{\sqrt{5} + \sqrt{7}} + . . . . + . . . .\] is
Sum of n terms of the series \[\sqrt{2} + \sqrt{8} + \sqrt{18} + \sqrt{32} +\] ....... is
The sum of the series 12 + 32 + 52 + ... to n terms is
The sum of the series \[\frac{2}{3} + \frac{8}{9} + \frac{26}{27} + \frac{80}{81} +\] to n terms is
If \[\sum^n_{r = 1} r = 55, \text{ find } \sum^n_{r = 1} r^3\] .
If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then write the value of k.
3 + 5 + 9 + 15 + 23 + ...
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 ______.
Let Sn(x) = `log_a 1/2 x + log_a 1/3 x + log_a 1/6 x + log_a 1/11 x + log_a 1/18 x + log_a 1/27x + ` ...... up to n-terms, where a > 1. If S24(x) = 1093 and S12(2x) = 265, then value of a is equal to ______.
The sum `sum_(k = 1)^20k 1/2^k` is equal to ______.
