Advertisements
Advertisements
Question
Show that `(1xx2^2 + 2xx3^2 + ...+nxx(n+1)^2)/(1^2 xx 2 + 2^2 xx3 + ... + n^2xx (n+1))` = `(3n + 5)/(3n + 1)`
Solution
nth term of the numerator = n(n + 1)2 = n3 + 2n2 + n
nth term of the denominator = n2(n + 1) = n3 + n2
APPEARS IN
RELATED QUESTIONS
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 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 3 × 8 + 6 × 11 + 9 × 14 +…
Find the sum to n terms of the series whose nth terms is given by (2n – 1)2
22 + 42 + 62 + 82 + ...
1 + (1 + 2) + (1 + 2 + 3) + (1 + 2 + 3 + 4) + ...
1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 + ...
Find the sum of the series whose nth term is:
n (n + 1) (n + 4)
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.
2 + 4 + 7 + 11 + 16 + ...
\[\frac{1}{1 . 6} + \frac{1}{6 . 11} + \frac{1}{11 . 14} + \frac{1}{14 . 19} + . . . + \frac{1}{(5n - 4) (5n + 1)}\]
If ∑ n = 210, then ∑ n2 =
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 + ...
Let Sn denote the sum of the cubes of first n natural numbers and sn denote the sum of first n natural numbers. Then, write the value of \[\sum^n_{r = 1} \frac{S_r}{s_r}\] .
The sum of the series
\[\frac{1}{\log_2 4} + \frac{1}{\log_4 4} + \frac{1}{\log_8 4} + . . . . + \frac{1}{\log_2^n 4}\] is
The sum of 10 terms of the series \[\sqrt{2} + \sqrt{6} + \sqrt{18} +\] .... is
Write the sum to n terms of a series whose rth term is r + 2r.
If \[\sum^n_{r = 1} r = 55, \text{ find } \sum^n_{r = 1} r^3\] .
3 + 5 + 9 + 15 + 23 + ...
2 + 5 + 10 + 17 + 26 + ...
Find the natural number a for which ` sum_(k = 1)^n f(a + k)` = 16(2n – 1), where the function f satisfies f(x + y) = f(x) . f(y) for all natural numbers x, y and further f(1) = 2.
Find the sum of the series (33 – 23) + (53 – 43) + (73 – 63) + … to n terms
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 ______.
The sum of all natural numbers 'n' such that 100 < n < 200 and H.C.F. (91, n) > 1 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 ______.
