Question

Tính tổng 

S=1+2+5+14+...+\(\frac{3^{n-1}+1}{2}\)

với n thuộc số nguyên dương

 

Answer 1

Đặt P=3^1-1+3^2-1+3^3-1+3^4-1+...+3^n-1

=>P=3⁰+3¹+3²+3³+...+3^n-1

=>3.P=3¹+3²+3³+3⁴+...+3ⁿ

=>3.P-P=3¹+3²+3³+3⁴+...+3ⁿ-3⁰-3¹-3²-3³-...-3^n-1

=>2.P=3ⁿ-3⁰

=>2.P=3ⁿ-1

=>\(P=\frac{3^n-1}{2}\)

Lại có: S=1+2+5+14+...+\(\frac{3^{n-1}+1}{2}\)

=>\(S=\frac{3^{1-1}+1}{2}+\frac{3^{2-1}+1}{2}+\frac{3^{3-1}+1}{2}+\frac{3^{4-1}+1}{2}+...+\frac{3^{n-1}+1}{2}\)

=>\(S=\frac{3^{1-1}+1+3^{2-1}+1+3^{3-1}+1+3^{4-1}+1+...+3^{n-1}+1}{2}\)

=>\(S=\frac{\left(3^{1-1}+3^{2-1}+3^{3-1}+3^{4-1}+...+3^{n-1}\right)+\left(1+1+1+1+...+1\right)}{2}\)

=>\(S=\frac{P+1.n}{2}\)

=>\(S=\frac{\frac{3^n-1}{2}+n}{2}\)

=>\(S=\frac{\frac{3^n-1}{2}+\frac{2n}{2}}{2}\)

=>\(S=\frac{\frac{3^n-1+2n}{2}}{2}\)

=>\(S=\frac{3^n-1+2n}{4}\)