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

To find: the value of r with respect to the binomial expansion of (1 + x)³⁴ where the coefficients of the (r – 5)th and (2r – 1)th terms are equal to each other

Formula Used:

The general term, T_r+1 of binomial expansionis given by,

T_r+1 ⁿC_r x^n-r y^r where

ⁿC_r

Now, finding the (r – 5)th term, we get

T_r-5³⁴C_r-6

Thus, the coefficient of (r – 5)th term is ³⁴C_r-6

Now, finding the (2r – 1)th term, we get

T_2r-1³⁴C_2r-2

Thus, coefficient of (2r – 1)th term is ³⁴C_2r-2

As the coefficients are equal, we get

³⁴C_2r-2³⁴C_r-6

2r-2=r-6

r=-4

Value of r=-4 is not possible

2r-2+r-6=34

3r=42

r=14

Thus, value of r is 14