GATE 2024 IN GA Q.4: solution and answer

GATE 2024 · INGA Q.4 · 1 mark · MCQ

If the sum of the first 20 consecutive positive odd numbers is divided by $2 0^{2}$ , the result is

(A)
1
(B)
20
(C)
2
(D)
1/2

See the question in the paper

Answer and solution

ANSWER

The sum of the first 20 positive odd numbers is $2 0^{2} = 400$ , so dividing by $2 0^{2}$ gives 1. The answer is option (A).

Official answer: ASource: Official answer key, GATE 2024 IN (IISc Bengaluru)

The first 20 consecutive positive odd numbers are $1, 3, 5, \dots, 39$ . The $n$ -th positive odd number is $2 n - 1$ , so the 20th is $2 (20) - 1 = 39$ .
These numbers form an arithmetic progression with common difference 2. Sum of an arithmetic progression = (number of terms) $\times$ (first term + last term) / 2: $S = \frac{20}{2} (1 + 39) = 10 \times 40 = 400$
This agrees with the standard result that the sum of the first $n$ positive odd numbers is $n^{2}$ : $S = 2 0^{2} = 400$ .
Divide by $2 0^{2}$ : $\frac{S}{2 0 ^{2}} = \frac{400}{400} = 1$
The result is 1, which is option (A).

KEY IDEA

The sum of the first $n$ positive odd numbers is always $n^{2}$ , because $1 + 3 + \dots + (2 n - 1) = \frac{n}{2} (1 + (2 n - 1)) = n^{2}$ . Here $n = 20$ , so the sum is exactly $2 0^{2}$ and the ratio is 1.

WATCH OUT FOR

Taking the 20th odd number to be 41 (confusing $2 n + 1$ with $2 n - 1$ ) or including 0, which gives a wrong sum. Another slip is using the formula for the sum of the first $n$ natural numbers, $\frac{n ( n + 1 )}{2} = 210$ , in place of the sum of odd numbers. Both lead to a ratio that is not 1.

From the GATE 2024 IN paper. Every question of this paper · Download the full paper · All 58 Numerical questions, with notes

GATE 2024 IN GA Q.4: solution and answer

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