Sum of Intermediate Terms in a Minimal Sumset Progression

Let A={1, a_1, a_2 … … a_18, 77} be a set of integers with 1

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Question type: Numerical
Exam relevance: JEE Main · Mathematics
Academic status: Answer verified against the official NTA key
Source: Previous-year question
Editorial review: 22 September 2026

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Why must the set be an arithmetic progression when |A + A| = 2|A| - 1?

Order the elements $x_{0} < x_{1} < \dots < x_{n - 1}$ . The chain $x_{0} + x_{0} < x_{0} + x_{1} < \dots < x_{0} + x_{n - 1} < x_{1} + x_{n - 1} < \dots < x_{n - 1} + x_{n - 1}$ contains exactly $2 n - 1$ strictly increasing values. If any gap between consecutive elements differed, intermediate sums would create extra distinct elements, forcing $∣ A + A ∣ > 2 n - 1$ .

Answer

Accepted answer: 702

The sum $a_{1} + a_{2} + \dots + a_{18}$ is equal to $702$ .

Why each option works or fails

Step-by-step solution

Given: The set is $A = {x_{0}, x_{1}, \dots, x_{19}} = {1, a_{1}, a_{2}, \dots, a_{18}, 77}$ with $∣ A ∣ = 20$ integers strictly increasing, and $∣ A + A ∣ = 39$ .
Goal: Find the sum of the intermediate elements: $S = a_{1} + a_{2} + \dots + a_{18}$ .
Approach: For any finite subset $A \subset Z$ with $∣ A ∣ = n$ , the minimal cardinality of the sumset $A + A$ is $2 n - 1$ . The equality $∣ A + A ∣ = 2 n - 1$ holds if and only if the elements of $A$ form an arithmetic progression. Here $n = 20$ , so $2 n - 1 = 39$ , meaning $A$ is in arithmetic progression.
Execute: Since $A$ forms an arithmetic progression with first term $1$ and $20^{th}$ term $77$ : $x_{19} = 1 + (20 - 1) d = 77 ⟹ 19 d = 76 ⟹ d = 4$ . Thus, the terms are $1, 5, 9, 13, \dots, 77$ . The required sum is $S = a_{1} + a_{2} + \dots + a_{18} = \sum_{i = 1}^{18} (1 + 4 i) = 18 \cdot 1 + 4 \cdot \frac{18 \cdot 19}{2} = 18 + 684 = 702$ .
Verify: Check using total sum: $\sum_{k = 0}^{19} x_{k} = \frac{20}{2} (1 + 77) = 10 \times 78 = 780$ . Subtracting endpoints $1$ and $77$ : $780 - 1 - 77 = 780 - 78 = 702$ .

Sum of Intermediate Terms in a Minimal Sumset Progression

What feels right?

The sum $a_{1} + a_{2} + \dots + a_{18}$ is equal to $702$ .

Why each option works or fails

Solution

01Given

02Goal

03Approach

04Execute

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