Working with geometric progressions

A geometric progression, or GP, is a sequence in which each term is found by multiplying the term before it by the same fixed number. That fixed number is the common ratio $r$.

In Add Math you reach for a GP to model anything that grows or decays by a constant factor rather than by a constant amount, money earning compound interest, a population that doubles each period, the value of a machine that falls to a fixed fraction of its worth each year, or the rebound heights of a bouncing ball that reach the same proportion of the previous drop.

The difference from an arithmetic progression matters: an arithmetic progression adds a fixed amount, so it grows in a straight line, while a geometric progression multiplies by a fixed factor, so it grows or shrinks ever faster. Three formulas cover almost every GP question.

The first finds any single term, the second adds up the first $n$ terms, and the third gives the total of an unending sum when the terms shrink towards zero.

The sum of the first $n$ terms also has a mirror-image form, $S_n = \frac{a(1-r^{n})}{1-r}$, obtained by multiplying the top and bottom of the first version by $-1$. The two are identical in value, but the second keeps every number positive when the common ratio is a proper fraction, so it is the tidier choice for a shrinking GP.

Knowing all three purposes at a glance, one term, a finite sum, an infinite sum, lets you match the formula to the question before you write a single number.

Reach for this method when a list of numbers is multiplied by the same factor at each step rather than having a fixed amount added. Test by dividing neighbouring terms: if $\frac{T_2}{T_1} = \frac{T_3}{T_2}$, the sequence is geometric and that shared value is $r$.

Words such as 'increases by a fixed percentage', 'doubles', 'halves', or 'each bounce reaches a fraction of the last' all point to a GP.

If the question mentions a total that keeps building forever while the terms get smaller, that is the signal for the sum to infinity, which is only valid when $|r| < 1$. Decide early whether you need a single term, a finite sum, or an infinite sum, because each needs a different formula and mixing them up is the most common way to lose marks on this topic.

A common exam dressing is percentage growth or decay. 'Increases by $20\%$ each year' means multiplying by $1.2$ every step, so $r = 1.2$; 'loses $20\%$ of its value each year' means multiplying by $0.8$, so $r = 0.8$.

Translating the percentage into a multiplier is the very first thing to do, because once you hold $r$ the rest of the question becomes routine substitution.

The common ratio need not be a whole number. It may be a fraction such as $\frac{1}{2}$ for a quantity that halves each step, or negative such as $-2$ for terms that swing between positive and negative signs.

A ratio lying strictly between $-1$ and $1$ is precisely the case where the sum to infinity exists, so judging the size of $r$ early also tells you whether a question is even entitled to ask for $S_\infty$. When in doubt, write out the first three or four terms; the pattern of multiplication is usually obvious once the numbers sit side by side.

1. 1

   Identify a and r

   Write down $a$ (the first term) and find $r$ by dividing any term by the one before it: $r = \frac{T_2}{T_1}$.

2. 2

   Set up equations if a or r is unknown

   If two terms are given, write each as $ar^{\,n-1}$, then divide one equation by the other so that $a$ cancels and only $r$ is left.

3. 3

   Solve for r, then a

   Solve the resulting equation for $r$, then substitute back into one term equation to find $a$.

4. 4

   Choose the right formula

   Use $T_n$ for a single term, $S_n$ for the sum of $n$ terms, and $S_\infty$ only when $|r| < 1$.

5. 5

   Substitute and simplify

   Put the numbers in carefully, working the powers out one line at a time, and state the final value.

6. 6

   Check

   List a few terms with your $a$ and $r$, or substitute back into a given term, to confirm the answer is sensible.

• Using $r^{n}$ instead of $r^{\,n-1}$ in the term formula, the first term needs zero multiplications by $r$, so the power is $(n-1)$.
• Adding instead of dividing when finding $r$; the common ratio comes from $\frac{T_2}{T_1}$, not $T_2 - T_1$.
• Applying the sum-to-infinity formula when $|r| \geq 1$, where the sum does not converge to a finite value.
• Losing a negative common ratio: if the terms alternate in sign, $r$ is negative and its powers change sign accordingly.
• Mixing up $S_n$ and $T_n$: a 'sum' question wants a total, a 'term' question wants a single value.

Most GP mistakes trace back to one moment, a student subtracts to find $r$ as though it were an arithmetic progression, or reaches for the sum to infinity when $|r|$ is not less than one. In one-to-one lessons our teachers watch exactly where you set up the ratio and the power, and correct the slip the instant it appears, so the wrong instinct never takes hold.

We also make you write the 'divide one equation by the other' line clearly, because that is where the common ratio is cleanly found and where method marks are earned. When a ratio turns out to be a fraction or a negative number, we slow down and rehearse the signs and powers until they feel natural.

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