Review question

# Can we find the sum of the integers from $2k$ to $4k$ inclusive? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R5849

## Suggestion

1. The positive integer $k$ is given.

1. Find, in terms of $k$, an expression for $S_1$, the sum of the integers from $2k$ to $4k$ inclusive.

2. Find, in terms of $k$, an expression for $S_2$, the sum of the odd integers lying between $2k$ and $4k$.

Is there a convenient way of dividing up these sums into smaller sums that are easier to deal with?

1. Prove that the sum of the first $n$ terms of the geometric progression having first term $a$ and common ratio $r$ ($r \neq 1$) is $a \left( \frac{1-r^n}{1-r} \right).$

Can we use the fact that the $k$th term in this geometric progression has the form $a\times r^{k-1}$?

If we call the sum of the first n terms $T_n$, do you notice anything about the terms in $r\times T_n$?

By regarding the recurring decimal $0.\dot{0}7\dot{5}$ ($=0.075075\dotsc$, where the figures $075$ repeat) as an infinite geometric progression, or otherwise, obtain the value of the decimal as a fraction in its lowest terms.

What’s the first term of the geometric progression? What’s the common ratio?