If the coefficients of `a^r-1`, `a^r` and `a^r+1` in the expansion of `(1+a)^n` are in arithmetic progression, prove that `n^2` - n(4r+1)+`4r^2` - 2 =0.

If the coefficients of (r-5)^th and (2r - 1)^th terms in the expansion of (1 + x)^34 are equal, find r.

The coeffcients of the (r-1)^th , r^th and (r+1)^th terms in the expansion of (x+1)^n are in the ration 1 : 3: 5 Find n and r.

Find the value of r , if the coefficients of (2r+4)th and (r-2) th terms in the expansion of (1+x)^(18) are equal .

Prove that the coefficient of x^n in the expansion of (1+x)^(2n) is twice the coefficient of x^n in the expansion of (1+x)^(2n-1) .

If the integers r gt 1 , n gt 2 and coefficients of (3)th and (r+2) nd terms in the Binomial expansion of (1+x)^(2n) are equal ,then

Prove that sum_(r=0)^n 3^r n Cundersetr = 4^n .

Two resistance R_(1) and R_(2) are made of different material. The temperature coefficient of the material of R_(1) is alpha and of the material of R_(2) is -beta . Then resistance of the series combination of R_(1) and R_(2) will not change with temperature, if R_(1)//R_(2) will not change with temperature if R_(1)//R_(2) equals

Let V_r denote the sum of the first r terms of an arithmetic progression (AP) whose first term is r and the common difference is (2r-1). Let T_r=V_(r+1) - V_r-2 and Q_r =T_(r+1) - T_r for r=1,2 T_r is always (A) an odd number (B) an even number (C) a prime number (D) a composite num,ber