Sum of First n Terms

Master Gauss's method and advanced sum formulas for arithmetic progressions

Sum Formula Mastery

Discover the elegant methods to find the sum of arithmetic progressions, from Gauss's brilliant insight to modern applications.

Sₙ = n/2[2a + (n-1)d]
Sₙ = n/2(a + l)
Gauss's Method
The Young Genius
When Gauss was 10, his teacher asked the class to find 1 + 2 + 3 + ... + 100. While others began adding slowly, Gauss found the answer in seconds: 5050.
Forward: S = 1 + 2 + 3 + ... + 98 + 99 + 100
1
2
3
...
98
99
100
Reverse: S = 100 + 99 + 98 + ... + 3 + 2 + 1
100
99
98
...
3
2
1
2S = 101 + 101 + 101 + ... + 101 (100 times)
S = 100 × 101 / 2 = 5050
Formula Derivation
General Method
Applying Gauss's technique to any arithmetic progression:
S = a + (a+d) + (a+2d) + ... + [a+(n-1)d]
S = [a+(n-1)d] + [a+(n-2)d] + ... + (a+d) + a
2S = [2a+(n-1)d] + [2a+(n-1)d] + ... + [2a+(n-1)d]
2S = n × [2a + (n-1)d]
S = n/2 × [2a + (n-1)d]
Alternative form using first and last terms:
Sₙ = n/2(a + l)
Worked Examples
Example 1: Direct Sum
Sum of first 22 terms: 8, 3, -2, ...
a = 8, d = -5, n = 22
S₂₂ = 22/2[2×8 + (22-1)×(-5)]
= 11[16 + 21×(-5)]
= 11[16 - 105] = -979
Example 2: Find Term from Sum
S₁₄ = 1050, a = 10, find 20th term
1050 = 14/2[20 + 13d]
1050 = 7[20 + 13d]
150 = 20 + 13d
d = 10, so a₂₀ = 10 + 19×10 = 200
Example 3: Find n from Sum
How many terms of 24, 21, 18... sum to 78?
a = 24, d = -3, Sₙ = 78
78 = n/2[48 + (n-1)(-3)]
156 = n[51 - 3n]
3n² - 51n + 156 = 0
n = 4 or n = 13
Sum Calculator
Calculate AP Sum
Enter the known values to calculate the sum:
Real-World Applications
Simple Interest
Investment of ₹1000 at 8% simple interest per year. Interest forms AP: 80, 160, 240, ...
30-year total interest:
S₃₀ = 30/2[2×80 + 29×80]
= 15[160 + 2320] = ₹37,200
Manufacturing Production
TV production: 600 sets in year 3, 700 in year 7. Find total production in first 7 years.
a = 550, d = 25
S₇ = 7/2[1100 + 6×25]
= 7/2[1250] = 4375 sets
Construction Penalty
Delay penalty: ₹200 first day, ₹250 second day, increasing by ₹50 daily. 30-day penalty?
a = 200, d = 50, n = 30
S₃₀ = 30/2[400 + 29×50]
= 15[1850] = ₹27,750
Explore Sum Concepts
Learning Progress
Master the sum formulas through historical context, mathematical derivation, and practical applications. From Gauss's brilliant insight to modern problem-solving techniques.