Geometric Progression

Chapter in a nutshell: A Geometric Progression (GP) is a sequence where each term is the previous one multiplied by a fixed common ratio $r$. You find the $n$th term, the sum of $n$ terms (different formula for $r>1$ and $r<1$), the sum to infinity (when $|r|<1$), and the geometric mean.

1. Key Ideas

• GP: $a,\ ar,\ ar^2,\ \dots$ — constant common ratio $r=\dfrac{a_2}{a_1}$.
• First term $a$, common ratio $r$, $n$th term $a_n$, number of terms $n$.

2. Formulas (quick reference — all of them)

• Common ratio: $\;r=\dfrac{a_{n}}{a_{n-1}}$
• $n$th term: $\;a_n=a\,r^{\,n-1}$
• Sum of $n$ terms:

• Sum to infinity (only if $|r|<1$): $\;S_\infty=\dfrac{a}{1-r}$
• Geometric mean of $a$ and $b$: $\;\text{GM}=\sqrt{ab}$
• Product/relation: the GM is the middle term of $a, \sqrt{ab}, b$ in GP.

3. Choosing Terms Cleverly (technique)

• 3 terms in GP: $\dfrac{a}{r},\ a,\ ar$ (product = $a^3$).
• 4 terms: $\dfrac{a}{r^3},\ \dfrac{a}{r},\ ar,\ ar^3$.

4. Prerequisite Formulas (earlier classes)

• Laws of indices: $a^m\cdot a^n=a^{m+n}$, $\dfrac{a^m}{a^n}=a^{m-n}$, $(a^m)^n=a^{mn}$, $a^0=1$.
• Square roots (for the geometric mean).

5. Worked Example (one, for the method)

6. Common Mistakes to Avoid

• Using the AP formulas by mistake (GP multiplies, AP adds).
• Using the wrong sum formula for the value of $r$ (>1 vs <1).
• Applying $S_\infty$ when $|r|\ge1$ (it only works for $|r|<1$).
• $a_n=a r^{n}$ instead of $a r^{\,n-1}$.
• GM as $\frac{a+b}{2}$ (that is the arithmetic mean; GM $=\sqrt{ab}$).

7. Likely Exam Questions (with crisp answers)

1. What is a GP? → A sequence with a constant common ratio.
2. Write the $n$th-term formula. → $a_n=ar^{\,n-1}$.
3. Write the sum formula for $r>1$. → $S_n=\frac{a(r^n-1)}{r-1}$.
4. When does a GP have a sum to infinity? → When $|r|<1$.
5. Write $S_\infty$. → $\frac{a}{1-r}$.
6. Geometric mean of a and b? → $\sqrt{ab}$.
7. How do you find $r$? → Divide any term by the previous one.
8. How are 3 numbers in GP usually taken? → $a/r,\ a,\ ar$.
9. GM of 4 and 9? → 6.
10. Difference between AP and GP? → AP has a common difference (add); GP has a common ratio (multiply).

Extended & Prerequisite Formulas (full)

• $a_n=ar^{n-1}$; $S_n=\tfrac{a(r^n-1)}{r-1}\,(r>1)=\tfrac{a(1-r^n)}{1-r}\,(r<1)$; $S_n=na\,(r=1)$; $S_\infty=\tfrac{a}{1-r}\,(|r|<1)$.
• GM of a,b = √(ab); 3 terms: a/r, a, ar.
• Laws of indices (prereq): $a^m a^n=a^{m+n}$; $a^m/a^n=a^{m-n}$; $(a^m)^n=a^{mn}$; $a^0=1$; $a^{-n}=1/a^n$.