Geometric Progressions and the AM-GM Inequality: Quick Revision

A brief note on Geometric Progressions and the AM-GM Inequality. Read it before a test. Try each question at the end before you open its answer.

The formulas

• $a$, $b$, $c$ are in GP exactly when $b^2 = ac$.
• Three numbers in GP: write them $\dfrac{a}{r}$, $a$, $ar$.
• $n$ means between $a$ and $b$: $r = \left(\dfrac{b}{a}\right)^{\frac{1}{n + 1}}$.
• For positive numbers, $A \ge G$, with equality only when they are equal.
• Two numbers with means $A$ and $G$ are the roots of $x^2 - 2Ax + G^2 = 0$.

A worked example

Find the sum of the first $6$ terms of $5, 15, 45, \ldots$, and the sum to infinity of $5 + \tfrac{5}{3} + \tfrac{5}{9} + \cdots$

Step 1. The first sum. $a = 5$ and $r = 3$: $\dfrac{5(3^6 - 1)}{3 - 1} = \dfrac{5 \times 728}{2} = 1820$.

Step 2. The second sum. $a = 5$ and $r = \tfrac{1}{3}$: $\dfrac{5}{1 - \tfrac{1}{3}} = \dfrac{15}{2}$.

Answer: $1820$ and $\tfrac{15}{2}$.

Practice with new numbers

A program to try

The program checks both sums of the worked example with exact fractions.

Mistakes to avoid

• $ar^n$ for the nth term. It is $ar^{n - 1}$.
• The sum formula with $r = 1$. The sum is $na$.
• A sum to infinity with $|r| \ge 1$. There is none.
• Forgetting the way up. A bouncing ball travels each rebound twice.
• AM-GM with negative numbers. It needs positive numbers.

Check yourself

1. Find the 4th term of $10, 5, \tfrac{5}{2}, \ldots$

2. Find the sum to infinity of $1 - \tfrac{1}{2} + \tfrac{1}{4} - \cdots$

3. Write $0.\overline{5}$ as a fraction.

4. Find the geometric mean of $2$ and $18$.

5. For $x > 0$, what is the least value of $x + \dfrac{25}{x}$?