SPI Mathematics · Sequence and Series

Geometric Progressions and Means

A geometric progression multiplies by a fixed common ratio r at every step; the arithmetic mean of two numbers is half their sum and the geometric mean is the square root of their product.

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Why this matters

Three questions in the bank, all MODERATE, and one is from the 2022 paper. They need two facts beyond the AP: a geometric progression multiplies instead of adding, and three numbers are in GP when the middle one squared equals the product of the other two.

Concept 1 of 2: Geometric progression: ratio, nth term and sum

In a geometric progression (GP) you multiply by the same number each time instead of adding it. That number is the common ratio. 5, 15, 45, 135 is a GP with common ratio 3.

Definition

Common ratio: $r = \dfrac{a_2}{a_1} = \dfrac{a_3}{a_2}$ .
nth term: $a_n = ar^{n-1}$ .
Sum of $n$ terms: $S_n = \dfrac{a(r^n - 1)}{r - 1}$ , for $r \ne 1$ .
Three numbers $a, b, c$ are in GP when $b^2 = ac$ .
A useful split: $S_{2n} = S_n(1 + r^n)$ , because the second $n$ terms are the first $n$ terms multiplied by $r^n$ .

Geometric progression

a_n = ar^{n-1} \qquad S_n = \frac{a(r^n - 1)}{r - 1} \qquad b^2 = ac

athe first term
rthe common ratio

Worked example

Find the 6th term and the sum of the first 6 terms of the GP 3, 6, 12, ...

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Sequence and Series · Geometric Progressions and Means

In G.P. sum of six terms = 126 and sum of three terms = 14 then its first term is

Using n instead of n minus 1 in the power

The 4th term of 5, 10, 20, ... is

5 \times 2^3 = 40

, not

5 \times 2^4 = 80

Testing a GP by subtracting

A GP is tested by dividing. 2, 4, 8 has equal ratios but unequal gaps, so it is a GP and not an AP.

Concept 2 of 2: Arithmetic and geometric means of two numbers

The arithmetic mean of two numbers fixes their sum, and the geometric mean fixes their product. Know both means and you know the two numbers.

Definition

Arithmetic mean: $\text{AM} = \dfrac{a + b}{2}$ , so the sum is $2 \times \text{AM}$ .
Geometric mean of two positive numbers: $\text{GM} = \sqrt{ab}$ , so the product is $\text{GM}^2$ .
Given the sum and the product, the numbers are the roots of $t^2 - (\text{sum})\,t + \text{product} = 0$ .
For positive numbers $\text{AM} \ge \text{GM}$ , with equality only when the two numbers are equal.
When a question mixes an AP and a GP, write the AP terms using $d$ , then apply $b^2 = ac$ to the GP.

The two means

\text{AM} = \frac{a + b}{2} \qquad \text{GM} = \sqrt{ab} \qquad t^2 - 2\,\text{AM}\,t + \text{GM}^2 = 0

Worked example

Two numbers have an arithmetic mean of 17 and a geometric mean of 15. Find them.

Practice this conceptself-check · 3 quick reps

The same idea in a real exam question:

SPI · 2022 · SPI Entrance Examination 2022 · Q27Moderate

Example 2 · Sequence and Series · Geometric Progressions and Means

If p, q and r are three unequal numbers such that p, q and r are in arithmetic progression and p, (r-q) and (q-p) are in geometric progression then p : q : r is equal to____.

Treating the geometric mean as the product

A geometric mean of 6 means the product is 36, not 6. Square the geometric mean before you use it.

Applying a ratio of means to the numbers

A ratio such as AM : GM = 5 : 4 compares the two means, not the two numbers. Find the AM from the sum first, then scale it to get the GM.

Drill 1 more on arithmetic and geometric means of two numbers

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Geometric progression: ratio, nth term and sum
Geometric progression
$a_n = ar^{n-1} \qquad S_n = \frac{a(r^n - 1)}{r - 1} \qquad b^2 = ac$
Arithmetic and geometric means of two numbers
The two means
$\text{AM} = \frac{a + b}{2} \qquad \text{GM} = \sqrt{ab} \qquad t^2 - 2\,\text{AM}\,t + \text{GM}^2 = 0$

Watch out for (4)

Using n instead of n minus 1 in the power
→ Geometric progression: ratio, nth term and sum
Testing a GP by subtracting
→ Geometric progression: ratio, nth term and sum
Treating the geometric mean as the product
→ Arithmetic and geometric means of two numbers
Applying a ratio of means to the numbers
→ Arithmetic and geometric means of two numbers

Test yourself on a real paper

Sit a past SPI paper, timed and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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