Question

For what value of m the coefficients of (2m+1)th and (4m+5)th terms, in the expansion of (1+x)th , are equal?

Answer 1

We know that the coefficient of mth term in the expansion of (1 + x)n is nCm - 1.

Therefore, coefficient of  (2m+1)th and (4m+5)th terms in the expansion of (1+x)10are 10C2m+1-1and 10C4m+5-1

It is given  that these coefficients are equal.

∴ 10C2m = 10C4m+4

⇒ 2m = 4m + 4 or 2m + 4m + 4 = 10  [As nCrand nCs ⇒ r = s or r + s = n]

⇒ -2m =  4 or 6m = 10 - 4  

⇒ m = -2 or 6m = 6

Now, m = -2 is not possible.  

Therefore, 6m = 6

⇒ m = 1

Answer 2

Answer:

Step-by-step explanation:

Since, the coefficient of mth term in the expansion of (1 + x)^n is n^Cm - 1.

Hence, coefficient of the (2m+1)th and (4m+5)th terms in the expansion of (1+x)^10are 10^C2m+1-1 and the second term is  10^C4m+5-1

∴ 10C2m = 10C4m+4

⇒ 2m = 4m + 4 or 2m + 4m + 4 = 10  [As nCrand nCk ⇒ r = k or r + k = n]

⇒ -2m =  4 or 6m = 10 - 4  

⇒ m = -2 or 6m = 6

Now, m = -2 is a negative value which is not possible

Therefore, 6m = 6

⇒ m = 1

Hope it helps :)