GATE 2018 EC GA Q.4: solution and answer

GATE 2018 · ECGA Q.4 · 1 mark · MCQ

What is the value of $1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \frac{1}{256} + \dots$ ?

(A)
$2$
(B)
$\frac{7}{4}$
(C)
$\frac{3}{2}$
(D)
$\frac{4}{3}$

See the question in the paper

Answer and solution

ANSWER

The series is geometric with $a = 1$ and $r = \frac{1}{4}$ , so its sum is $\frac{1}{1 - \frac{1}{4}} = \frac{4}{3}$ . The answer is option (D).

Official answer: DSource: Official answer key, GATE 2018 EC (IIT Guwahati)

Each term is $\frac{1}{4}$ of the one before it: $\frac{1}{4} \div 1 = \frac{1}{4}$ , $\frac{1}{16} \div \frac{1}{4} = \frac{1}{4}$ , $\frac{1}{64} \div \frac{1}{16} = \frac{1}{4}$ , and so on. So this is a geometric series with first term $a = 1$ and common ratio $r = \frac{1}{4}$ .
An infinite geometric series converges when $∣ r ∣ < 1$ , and its sum is then $S_{\infty} = \frac{a}{1 - r}$ . Here $∣ r ∣ = \frac{1}{4} < 1$ , so the sum exists.
$S_{\infty} = \frac{1}{1 - \frac{1}{4}} = \frac{1}{\frac{3}{4}} = \frac{4}{3}$
Check with partial sums: $1 + 0.25 + 0.0625 + 0.015625 + 0.00390625 = 1.33203125$ , which is closing in on $1.333 \dots = \frac{4}{3}$ and stays well below $\frac{3}{2} = 1.5$ .
So the value is $\frac{4}{3}$ , which is option (D).

KEY IDEA

If every term is a fixed multiple $r$ of the one before it and $∣ r ∣ < 1$ , the infinite geometric series adds up to $\frac{a}{1 - r}$ , where $a$ is the first term. Find $a$ and $r$ first, then use the formula.

WATCH OUT FOR

Using the ratio $\frac{1}{2}$ because the series looks like the familiar $1 + \frac{1}{2} + \frac{1}{4} + \dots = 2$ . That gives option (A). Another slip is to leave out the leading $1$ and add only $\frac{1}{4} + \frac{1}{16} + \dots = \frac{1}{3}$ . Adding the first few terms also shows the answer is below $1.34$ , which rules out (A), (B) and (C).

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GATE 2018 EC GA Q.4: solution and answer

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