Q.32 A series (S) is given as

S = 1 + 3 + 5 + 7 + 9 + ……

The sum of the first 50 terms of S is ____________.

Sum of First 50 Odd Numbers

The series S = 1 + 3 + 5 + 7 + 9 + … represents the sum of the first 50 odd numbers.
The total sum is 2500.

Series Identification

This is an arithmetic progression (AP) with:

• First term a = 1
• Common difference d = 2

The nth term is:

an = a + (n − 1)d = 2n − 1

The 50th term is:

2 × 50 − 1 = 99

Step-by-Step Calculation

Sum of first 50 terms using AP formula:

S₅₀ = n/2 [2a + (n − 1)d]
S₅₀ = 50/2 [2 × 1 + 49 × 2]
     = 25 [2 + 98]
     = 25 × 100
     = 2500

Alternatively, using:

Sₙ = n/2 (a + l)

with last term l = 99:

50/2 (1 + 99) = 25 × 100 = 2500

Matches the identity for first n odd numbers:

Sₙ = n² = 50² = 2500

Options Explanation

Common incorrect answers include:

• 25000 — misreading n = 500
• 5200 or 6250 — wrong formula arithmetic
• 20500 — miscomputed last term

Correct Answer: 2500

The sum of first 50 odd number terms equals n². For n = 50, sum = 2500.

Why It’s an Arithmetic Progression

General term aₙ = 2n − 1, so:

• 1st term = 1
• 2nd = 3
• ⋯
• 50th = 99

Common difference d = 2.

Formula and Proof

Use AP sum formulas:

• Sₙ = n/2 [2a + (n − 1)d]
• Sₙ = n² (shortcut for odd numbers)

Therefore: S₅₀ = 2500.