Arithmetic and Geometric Sequences

Try this

Drag “pair $i$ ”: are the three readouts $a_{1} + a_{6}$ , $a_{2} + a_{5}$ , $a_{3} + a_{4}$ the same? Change the first term and the common difference and try again.
Press “Stack reversed”: how tall does each column become? How many columns do the two copies fill?
Make $n$ odd. The middle term can only pair with itself. How is it related to $a_{1} + a_{n}$ ?
In general, if $m + n = p + q$ , then $a_{m} + a_{n} = a_{p} + a_{q}$ . Why? (Hint: $a_{k} = a_{1} + (k - 1) d$ is a linear function of $k$ .)

Write $S_{n}$ forwards once and backwards once and add the two rows: each column is $a_{1} + a_{n}$ , and there are $n$ columns:

2 S_{n} = n (a_{1} + a_{n}), S_{n} = \frac{n ( a _{1} + a _{n} )}{2} = n a_{1} + \frac{n ( n - 1 )}{2} d .

This is adding in reverse order. In the picture, the two staircases fit together into a neat rectangle.

Predict first

In an arithmetic sequence, $a_{3} + a_{8} = 20$ . Find the sum of the first $10$ terms, $S_{10}$ .

Choose an answer first; then you can see the result.

Why: shift and subtract for geometric sequences

In a geometric sequence $a_{k} = a_{1} q^{k - 1}$ , each term is $q$ times the one before. So if you multiply the whole of $S_{n}$ by $q$ , you get almost the same terms back, just shifted one place to the right.

Example: 3 + 6 + 12 + 24 + 48

Step 1 / 5

Write down S
First term $3$ , common ratio $2$ , $5$ terms. The top row is $S$ .

$S = 3 + 6 + 12 + 24 + 48$
Multiply by q
Subtract
The general formula
Pairing works too

Arithmetic and geometric sequences are a “dual” pair: an arithmetic sequence is a linear function of $k$ and behaves well under addition; a geometric sequence is an exponential function of $k$ and behaves well under multiplication. Take logarithms, and a geometric sequence becomes an arithmetic one.

Remove one condition

Break it

Counterexample lab: remove one condition

Conclusion: $S_{n} = \frac{a _{1} ( q ^{n} - 1 )}{q - 1}$

$q = 3$ : press “Shift and subtract”, and the formula readout agrees exactly with the term-by-term sum.

Practice

Exercise 1Basic

In an arithmetic sequence, $a_{4} = 10$ and $a_{9} = 25$ . Find $a_{20}$ .

Exercise 2Basic

Find $1 + 3 + 3^{2} + \dots + 3^{7}$ .

Exercise 3Advanced

Find $1 \cdot 2 + 2 \cdot 2^{2} + 3 \cdot 2^{3} + \dots + 6 \cdot 2^{6}$ . (Arithmetic times geometric: shift-and-subtract still works.)

Exercise 4Challenge

The sum of the first $5$ terms of an arithmetic sequence is $25$ , and the sum of the first $10$ terms is $100$ . Find the sum of the first $15$ terms.

Try this

Why: shift and subtract for geometric sequences

Example: 3 + 6 + 12 + 24 + 48

Remove one condition

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Treating aₘ + aₙ as aₘ₊ₙ