Sums of sequences

The story goes that a young Gauss added 1 to 100 in seconds. Pair the ends: $1 + 100$ , $2 + 99$ and so on. That's 50 pairs of 101, so 5050.

1 + 2 + \dots + 6

pairs into 3 sevens: 21.

Work through it

Count the terms. $d = 4$ . $n = 10$

Average the ends and multiply. $\frac{10}{2} (3 + 39) = 210$

It's geometric with $r = \frac{1}{2}$ , so it converges. $\frac{1}{1 - 1/2}$

Calculate. $2$

Using the infinite formula with $r = 2$ .

It only works when $∣ r ∣ < 1$ . Otherwise the sum keeps growing.

The short version
Arithmetic:

S_{n} = \frac{n}{2} (a_{1} + a_{n})

. Geometric:

S_{n} = a_{1} \frac{1 - r ^{n}}{1 - r}

. Infinite geometric, with

∣ r ∣ < 1

\frac{a _{1}}{1 - r}

Still stuck? Talk it through with a tutor. The first consultation is free. See our Precalculus tutoring.