Mathematics

Problem

Consider the infinite geometric sequence

160, 80\sqrt{2}, 80, 40\sqrt{2}, \dots

a. Write the 12th term of the sequence in the form

k\sqrt{2}

where

k \in \mathbb{Q}

. b. Find, in the form

a + b\sqrt{2}

where

a, b \in \mathbb{Z}

: - i**

S_{10}

- ii** the sum

S

of the infinite series.