Geometric Progressions: Finite and Infinite Sums

1. From Common Difference to Common Ratio: What is a GP?

A Geometric Progression (GP) is a sequence where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

• In an AP, we add/subtract: 2, 5, 8, 11... (common difference $d=3$)
• In a GP, we multiply/divide: 2, 6, 18, 54... (common ratio $r=3$)

Let's denote the first term by $a$ and the common ratio by $r$. The sequence looks like this:
$a,ar,a{r}^2,a{r}^3,\dots$

This pattern of multiplicative growth is something you've likely encountered in computer science, for instance when analyzing algorithms with complexities like $O(2^n)$ where the problem size doubles at each recursive step.

Finding the nth Term

Just as with APs, we can find a formula for any term in the sequence.

• 1st term ($t_1$) = $a=a\cdot r^0$
• 2nd term ($t_2$) = $a\cdot r=a\cdot r^1$
• 3rd term ($t_3$) = $a\cdot r^2$
• 4th term ($t_4$) = $a\cdot r^3$

The power of $r$ is always one less than the term number. This gives us the formula for the nth term:

$t_n=a\cdot r^{(n-1)}$

To solidify these foundational concepts, let's watch a short, clear explanation.

2. The Sum of a Geometric Progression

Now, let's tackle the sum of the terms in a GP. There are two main scenarios: summing a finite number of terms and summing an infinite series.

Sum of a Finite GP

The formula for the sum of the first $n$ terms, $S_n$, depends on whether $r$ is greater or less than 1. This is just to keep the denominator positive, but both forms are mathematically equivalent.

• If $r>1$, use: $S_n=\frac{a(r^n-1)}{r-1}$
• If $r<1$, use: $S_n=\frac{a(1-r^n)}{1-r}$
• If $r=1$, the series is just $a,a,a,\dots$ and $S_n=na$.

To understand where these formulas come from, which is key to avoiding confusion during an exam, let's see a quick derivation.

Sum of an Infinite GP

This is a fascinating and frequently tested concept. Can we sum a series that goes on forever? Yes, but only under one crucial condition: the common ratio $r$ must be between -1 and 1 (i.e., $|r|<1$).

Why? If $r$ is a fraction like 1/2, each term gets progressively smaller (e.g., 8, 4, 2, 1, 0.5, ...), eventually becoming so small that the sum approaches a finite limit. If $|r|\ge 1$, the terms either grow or stay the same size, and the sum would be infinite.

The formula is elegantly simple:
$S_{\infty }=\frac{a}{1-r}(\text{only if }-1<r<1)$

For a quick reference on all the sum formulas and the condition for the infinite sum, let's consult a text resource.

The following flowchart is an excellent summary to help you decide which formula to use in any given problem.

3. CAT-Specific Problems and Techniques

For the CAT, simply knowing the formulas isn't enough. You need to recognize problem patterns and apply efficient techniques.

Technique 1: Choosing Terms Strategically

If a problem gives you the product of three terms in a GP, don't assume them as $a,ar,a{r}^2$. Instead, choose them as:
$\frac{a}{r},a,ar$
When you multiply these, the $r$'s cancel out, leaving you with $a^3$. This simplifies the problem immensely. The middle term is now 'a'.

The following video demonstrates this powerful technique.

Technique 2: The Bouncing Ball Problem

This is a classic application of the infinite GP sum. A ball is dropped from a height $H$ and rebounds to a fraction $r$ of its previous height. The total distance traveled is an infinite GP.

• Initial Drop: $H$
• First Rebound (up and down): $Hr+Hr=2Hr$
• Second Rebound (up and down): $H{r}^2+H{r}^2=2H{r}^2$
• ...and so on.

The total distance is $H+2Hr+2H{r}^2+2H{r}^3+\dots$
This is $H$ plus an infinite GP with first term $2Hr$ and common ratio $r$.
Total Distance = $H+S_{\infty }=H+\frac{2Hr}{1-r}$.

Let's watch a masterclass on solving this exact problem type.

Conclusion

Today we've unpacked Geometric Progressions, a vital topic for the CAT. You've learned to identify a GP, find any term, and calculate its sum, both for finite and infinite series. Most importantly, you've seen how these concepts are applied in exam-style questions.

Key Takeaways:

• A Geometric Progression (GP) has a constant ratio ($r$) between terms.
• The nth term is given by $t_n=a\cdot r^{(n-1)}$.
• The sum of a finite GP is $S_n=\frac{a(r^n-1)}{r-1}$.
• The sum of an infinite GP exists only if $|r|<1$, and the formula is $S_{\infty }=\frac{a}{1-r}$.
• For problems involving the product of terms, strategically choose them as $\frac{a}{r},a,ar$.
• Recognize the "bouncing ball" problem as a classic infinite GP application.

Preview of the Next Lesson:
We will round out our study of fundamental series by learning how to apply formulas for the sum of first n natural numbers, their squares, and their cubes. These are special, fixed formulas that are incredibly useful as building blocks for solving more complex summation problems.