Question

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

Answer 1

Given expression is `(p/2 + 2)^8` 

Number of terms = 8 + 1 = 9 (odd)

∴ Middle term = `(9 + 1)/2` th term = 5^th term

∴ T₅ = T₄₊₁

= `""^8"C"_4 (p/2)^(8 - 4)  (2)^4`

= `""^8"C"_4  p^4/2^4 xx 2^4`

= `""^8"C"_4 p^4`

Now ⁸C₄P⁴ = 1120

⇒ `(8 xx 7 xx 6 xx 5)/(4 xx 3 xx 2 xx 1) "P"^4` = 1120

⇒ 70P⁴ = 1120

⇒ P⁴ = `1120/70` = 16

⇒ P⁴ = 2⁴

⇒ P = ±2

Hence, the required value of P = ±2