## Arithmetic Sequences

An arithmetic sequence is a sequence of the form $$ a, a+d, a+ 2d, a+3d, \dots $$ where a and d are constant. The $n^{th}$ term, often denoted U_n is given by $$ U_n = a + (n-1)d $$ The sum of the first n terms is given by $$ S_n = \frac{n}{2}(2a + (n-1)d) $$ This is often proven by considering the $2S_n$, written as $$2S_n = (U_1 + U_n) + (U_2 + U_{n-1}) + \dots (U_n + U_1)$$ We note that $U_i + U_n-i = 2a + (n-1)d$ for all $I$. Therefore $$ 2S_n = n(2a + (n-1)d) $$ which gives our desired formula $$ S_n = \frac{n}{2}(2a + (n-1)d) $$

## Arithmetic Sequences 1

A simple saving plan involves saving £10 at the start of month 1, £12 at the start of month 2, £14 at the start of month 3 and so on. Determine the amount invested at the start of month 24 and the total amount invested over the period.

solution - press button to display

The sequence is an arithmetic sequence as the increase is constant. $$U_n = a + (n-1)d$$ $a$ represents $U_1$, which in this case corresponds to month 1. The investment at the start of month 24 is given by $U_{24}$, with $a=10$ and $d=2$. $$ U_{24} = 10 + 23(2) = £56 $$ The total amount invested is given by $S_{24}$ $$ S_n = \frac{n}{2}\left(2a + (n-1)d\right) = \frac{24}{2}\left(20 + 23\times 2\right) = 792 $$

## Arithmetic Sequences 2

An arithmetic sequence begins $100, 97, 94, \dots$. Determine the value of $n\gt 0$ for which $S_n$ is maximal. State also the value of $S_n$

solution - press button to display

For this sequence, $a=100,\;d=-3$. $S_n$ has its maximal value around when $U_n = 0$. $$ U_n = 100 + (n-1)(-3) = 103 - 3n $$ $U_n = 0 \Rightarrow 103 = 3n \Rightarrow n = 34\frac{1}{3}$ It follows that $U_{35} \lt 0$, therefore $S_{34} \gt S_{35}$ so $S_{34}$ is maximal. The value of $S_{34}$ is $$ S_{34} = \frac{34}{2}\left(2(100) + (34-1)(-3)\right) = 1717 $$