Question

Question 18 Find the first four partial sums and the nth partial sum of the sequence an 2 ܚ | ܀ 4 Give your answers as fractions. Si S2= S3- S4 - S.

Answer 1

3 4

$\small S_{1}=\frac{3}{4^{1}}=\frac{3}{4}$

$\small S_{2}=S_{1}+a_{2}=\frac{3}{4}+\frac{3}{16}=\frac{15}{16}$

$\small S_{3}=S_{2}+a_{3}=\frac{15}{16}+\frac{3}{64}=\frac{63}{64}$

$\small S_{4}=S_{3}+a_{4}=\frac{63}{64}+\frac{3}{256}=\frac{255}{256}$

If we look at pattern, we can comclude:

i) Numerator is one less than denominator,

ii) Denominator is following 4ⁿ pattern.

$\small \therefore S_{n}=\frac{4^{n}-1}{4^{n}}$