The sum of infinity of the G.P. a, ar, ar², ar³, ...... \(\infty\) is ______.

The seen to infinity of the series \(1+2.\frac{1}{2}+3.\frac{1}{2^2}+4.\frac{1}{2^3}+...+\infty\) ______.

The sum of the square of (n - 1) natural numbers is ______.

The sum to infinity of \(\frac{1}{3}+\frac{3}{9}+\frac{5}{27}+\frac{7}{81}+...+\infty\) is ______.

Sum of n terms of ,the series \(\sqrt { 2 } +\sqrt { 8 } +\sqrt { 18 } +\sqrt { 32 } +\)..... is ______.

If the sum of first n even natural numbers is equal to m times the sum of first n is odd natural numbers then m is equal to ______.

In a G.P. if the (m + n)^th terms is p and (m - n)^th terms is q then its mth terms is ______.

The three geometric means between the numbers 1 and 81 are ______.

If a, b, c are in G.P. and \({ a }^{ \frac { 1 }{ x } }={ b }^{ \frac { 1 }{ y } }={ c }^{ \frac { 1 }{ z } }\), then xyz are in ______.

If the first term of an AP. is 5 and common difference is - 3 then sum of its 60 terms is equal to ______.

If the first, 'second and last term of an AP. are a, band 2a respectively then its sum is equal to ______.

If Sn denote the sum of n terms of an  AP. whose first term is a and common difference is d given by is d given by d = S_n - kS_n-1 + S_n-2 then K is equal to ______.

If the sum of n terms of an AP. is 4n² + 7n, then its nth term is ______.

If first and last terms ofanAP. are 3 and 18 and the sum of its terms is 84, then number of terms will be ______.