If (2+ x 3 )⁵⁵ is expanded in the ascending powers of x and the coefficients of powers of x in two consecutive terms of the expansion are equal, then these terms are:

Step-by-step solution

Let r^ th and (r+1)^ th term has equal coefficient aligned & (2+ x 3 )⁵⁵=2⁵⁵ (1+ x 6 )⁵⁵ &r^ th term =2⁵⁵⁵⁵ C_r ( x 6 )^r aligned Coefficient of x^r is 2⁵⁵ ⁵⁵ C_r 1 6^r (r+1)^ th term =2⁵⁵⁵⁵ C_ r+1 ( x 6 )^ r+1 Coefficient of x^ r+1 is 2⁵⁵ 55 r+1 1 6^ r+1 Both coefficients are equal 2⁵⁵⁵⁵ C_r 1 6^r =2⁵⁵⁵⁵ C_ r+1 1 6^ r+1 aligned & 1 |r| 55-r = 1 r+1 54-r 1 6 &6(r+1)=55-r &6 r+6=55-r &7 r=49 &r=7 &(r+1)=8 aligned Coefficient of 7^ th and 8^ th terms are equal.