Geometric sequences.

Step 1 of 5

Multiply by the ratio

In a geometric sequence each term is the one before multiplied by the same number, the ratio $q$⁠: $a_{n+1} = a_n \cdot q$$a_{n+1} =$$a_n \cdot q$⁠. This is exponential growth from the exponents Unit, written as a list. On the board it is the staircase on the right. Here $a_1 = 1$⁠ and Ratio q is $2$⁠. Predict the first six terms.

Step 2 of 5

The general term and the ratio

From the first term to the $n$⁠th there are $n - 1$⁠ steps, and each step multiplies by $q$⁠:

$a_n = a_1 q^{n-1}$⁠

Here the first term is $a_1 = 4$⁠, and Terms n is locked at $3$⁠. Find the ratio that gives $a_3 = 9$⁠. A hint from the exponents Unit: two steps of multiplying by $q$⁠ are a multiplication by $q^2$⁠.

Step 3 of 5

A sum of powers of 2

The sum of a geometric sequence takes a different trick. Write the sum, $S_n = a_1 + a_1 q + \dots + a_1 q^{n-1}$$S_n =$$a_1 + a_1 q + \dots + a_1 q^{n-1}$⁠, and under it $q S_n$⁠: the same terms, each moved along one place. Subtract, and almost everything cancels, leaving $q S_n - S_n = a_1 q^n - a_1$$q S_n - S_n =$$a_1 q^n - a_1$⁠:

$S_n = \frac{a_1(q^n - 1)}{q - 1}$$S_n =$$\frac{a_1(q^n - 1)}{q - 1}$⁠

The formula works for any $q \neq 1$⁠. Here $q = 2$⁠: work out $S_{10}$⁠, then move Add the terms to the running totals.

Step 4 of 5

A sum of halves

When $q < 1$⁠ the terms shrink, and it is easier to write the same formula with both signs turned round: $S_n = \frac{a_1(1 - q^n)}{1 - q}$$S_n =$$\frac{a_1(1 - q^n)}{1 - q}$⁠. Here $q = 0.5$⁠, each term half the one before. Work out the sum of the first six terms.

Step 5 of 5

Which one wins

The board now shows both sequences, both from $a_1 = 1$⁠: on the left you add $3$⁠, on the right you multiply by $1.5$⁠. Over the first four terms the arithmetic one is well ahead. Predict which will be bigger at the tenth term, then move Terms n and check.