If p is a real number and the middle term in the expansion of `((p)/(2) + 2)^(8)` is 1120, then find the value of p.

Explanation

Given expansion is `((p)/(2) + 2)^(8)`
Hence, `n = 8`
Since, this Binomial expansion has only one middle term i.e., `((8)/(2) + 1)th = 5 th` term
`T_(5) = T_(4 + 1) = .^(8)C_(4) ((p)/(2))^(8 - 4) .2^(4)`
`rArr 1120 = .^(8)C_(4) p^(4) .2^(-4) 2^(4)`
`rArr 1120 = (8 xx 7 xx 6 xx 5 xx 4!)/(4! xx 4 xx 3 xx 2 xx 1)p^(4)`
`rArr 1120 = 7 xx 2 xx 5 xx p^(4)`
`rArr p^(4) = (1120)/(70) = 16 rArr p^(4) = 2^(4)`
`rArr p^(2) = 4 rArr p = -+ 2`