Eduprobe
Question:
If 1+x⁴+x⁵=∑⁵ᵢ₌₀ aᵢ(1+x)ⁱ, for all x in R, then a₄ is:
-4
6
-8
10
Solution:
1+x⁴+x⁵=a₀+a₁(1+x)+a₂(1+x)²+a₃(1+x)³+a₄(1+x)⁴+a₅(1+x)⁵
Comparing x⁵ terms, we get a₅=1
Coefficient of x⁴ in RHS is a₄+5a₅
So, 1=a₄+5. a₄=-4
-4
6
-8
10
1+x⁴+x⁵=a₀+a₁(1+x)+a₂(1+x)²+a₃(1+x)³+a₄(1+x)⁴+a₅(1+x)⁵
Comparing x⁵ terms, we get a₅=1
Coefficient of x⁴ in RHS is a₄+5a₅
So, 1=a₄+5. a₄=-4