A pendulum swings through an arc of 25 cm. On each successive swing, the pendulum covers an arc equal to 90% o

Worked Examples · Example 36

Q. A pendulum swings through an arc of 25 cm. On each successive swing, the pendulum covers an arc equal to 90% of the previous swing. Find the length of the arc on the sixth swing and the total distance the pendulum travels before coming to rest.

CBSENCERTSubjective· 3mImportance★★★★★est

96% · 102/106 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Infinite Geometric Progression

Infinite Geometric Progression

The intuition: what happens when you keep halving?

Imagine you start walking toward a wall. In the first step, you cover half the remaining distance. In the next step, you cover half of what's left. Then half of that. And so on.

The distances you cover in each step are:

$\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}, \dots$

This is a geometric progression — each term is the previous term multiplied by a fixed number (here, $\frac{1}{2}$ ). That fixed multiplier is called the common ratio, $r$ .

Now ask: if you keep doing this forever, what is the total distance you cover? Intuition says you never quite reach the wall — but you get arbitrarily close. The total distance covered after $n$ steps is:

$S_n = \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \dots + \frac{1}{2^n}$

This sum gets closer and closer to $1$ as $n$ grows. It never exceeds $1$ , but it approaches $1$ as closely as you like. We say the infinite sum equals $1$ .

That is the core idea of an infinite geometric progression: a GP that goes on forever, and whose sum (under the right condition) settles to a finite number.

The precise statement

An infinite geometric progression is a sequence of the form:

$a, ar, ar^2, ar^3, ar^4, \dots$

where $a$ is the first term and $r$ is the common ratio. The sequence never ends.

The sum of the first $n$ terms (the finite sum) is:

$S_n = a \frac{1 - r^n}{1 - r}, \quad r \neq 1$

Now, what happens as $n \to \infty$ ? That depends entirely on $r$ .

Important

The infinite sum $S_\infty = a + ar + ar^2 + \dots$ exists only when $|r| < 1$ . In that case:

$S_\infty = \frac{a}{1 - r}$

If $|r| \geq 1$ , the sum either blows up to infinity or oscillates without settling — it does not have a finite value.

Why $|r| < 1$ is the key

Look at $S_n = a \frac{1 - r^n}{1 - r}$ . The only part that depends on $n$ is $r^n$ .

If $|r| < 1$ , then $r^n \to 0$ as $n \to \infty$ . So $S_n \to \frac{a}{1 - r}$ .
If $|r| > 1$ , then $r^n$ grows without bound — the sum diverges.
If $r = 1$ , the formula breaks (division by zero), and the sum is $a + a + a + \dots$ , which diverges. …

Start your 14-day free trial to unlock the full solution →

Each swing's arc is $90\%$ of the previous one, so the arc lengths form a geometric progression; the $6^{\text{th}}$ term gives the sixth-swing arc, and the infinite sum gives the total distance before the pendulum stops.

a

n\text{th term: } a_n = ar^{n-1}, \qquad \text{Sum to infinity }(|r|<1): S_\infty = \frac{a}{1-r}

Identify the GP: first swing arc $a = 25$ cm; each successive swing is $90\%$ of the previous, so common ratio $r = 0.9$ .
Arc on the $6^{\text{th}}$ swing uses $a_6 = ar^{6-1} = ar^5$ :

a_6 = 25 \times (0.9)^5

Compute $(0.9)^5$ step by step: $(0.9)^2=0.81$ , $(0.9)^3=0.729$ , $(0.9)^4 = 0.6561$ , $(0.9)^5 = 0.59049$ .
$a_6 = 25 \times 0.59049 = 14.76225 \approx 14.76 \text{ cm}$
Total distance travelled before the pendulum comes to rest is the sum of ALL swing arcs, out to infinity (since $r=0.9$ , $|r|<1$ , the series converges): …

Start your 14-day free trial to unlock the full solution →

←Previous question · 5.6Example 35 — A substance initially weighing 64 grams is decaying at a rate such tha…→Next question · 5.6Example 37 — A square is drawn by joining the mid-points of the sides of a square. …

Q.A pendulum swings through an arc of 25 cm. On each successive swing, the pendulum covers an arc equal to 90% of the previous swing. Find the length of the arc on the sixth swing and the total distance the pendulum travels before coming to rest.

🎓Unlock everything free for 14 days

Unlock everything free for 14 days