Question

(FGV-SP)A soma dos coeficientes do desenvolvimento de (2x+y)^5 é igual a?

A)81
B)128
C)243
D)512
E)729

Answer 1

$(2x+y)^5$

Em geral,

$(a+b)^{n}=\dbinom{n}{0}a^{n}b^0+\dbinom{n}{1}a^{n-1}b^1+\dots+\dbinom{n}{n-1}a^{1}b^{n-1}+\dbinom{n}{n}a^0b^{n}$

Assim:

$(2x+y)^5=\dbinom{5}{0}(2x)^5y^0+\dbinom{5}{1}(2x)^4y^1+\dots+\dbinom{5}{4}(2x)^1y^4+\dbinom{5}{5}(2x)^0y^5$

A soma dos coeficientes de $(2x+y)^5$ é:

$\dbinom{5}{0}2^5+\dbinom{5}{1}2^4+\dbinom{5}{2}2^3+\dbinom{5}{3}2^2+\dbinom{5}{4}2^1+\dbinom{5}{5}2^0$

$=2^5+5\cdot2^4+10\cdot2^3+10\cdot2^2+5\cdot2+1$

$=32+80+80+40+10+1=243$

Alternativa C