Jun 4, 2024
Given Series: 1, 2, 3, ..., 100.
Method: Write the series in reverse and add it to the original series.
S = 1 + 2 + 3 + ... + 100
S = 100 + 99 + 98 + ... + 1
(1+100) + (2+99) + (3+98) + ...
Calculation:
S + S = 100 * 101
2S = 100 * 101
S = 100 * 101 / 2 = 5050
Define Variables:
a = first term
d = common difference
n = number of terms
Series Representation:
Sₙ = a + (a + d) + (a + 2d) + ... + [a + (n-1)d]
Use the same summing trick:
Sₙ = a + (a + d) + (a + 2d) + ... + [a + (n-1)d]
Sₙ = [a + (n-1)d] + [a + (n-2)d] + ... + a
2a + (n-1)d
n
Derive the Formula:
2Sₙ = n * [2a + (n-1)d]
Sₙ = n/2 * [2a + (n-1)d]
Sum of Arithmetic Series:
[ Sₙ = \frac{n}{2} [2a + (n-1)d] ]