These look long but each has a one-line formula. Never list everything out by hand.
Formulas that turn long sums into one line
• Sum of first n numbers = 2n(n+1)
• Sum of their squares = 6n(n+1)(2n+1)
• Sum of their cubes = [2n(n+1)]2 (always a perfect square!)
• Count from a to b divisible by d = ⌊db⌋−⌊da−1⌋
Consecutive numbers: for an odd count of equally spaced numbers, the average is the middle number. Smallest = average − (n−1), largest = average + (n−1).
📊Series Sum BuilderAdd 1…n the fast way
n = 6
Sum 1+2+…+6 = n(n+1)/2 = 6×7/2 = 21
Total = 21
Sum of the first 10 natural numbers = 10 × 11 ÷ 2 = .
📝Practice Questions
Q1The sum of 5 consecutive numbers is 100. The largest is:
Q2How many numbers from 1 to 100 are divisible by 4?
Q3The sum of the first 15 natural numbers is:
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Real exam questions — Patterns & Counting
3 question types · 9 solved examples from real SSC papers
Consecutive numbers, counting in a range, and series sums — each has a one-line formula that beats listing everything out.
How to solve this type
Write the terms as x, x+1, x+2 (use step 2 for even or odd numbers) and add them; the constants give a quick linear equation. For an odd count of equally spaced terms the average equals the middle term, so middle = sum ÷ count. Handy shortcuts: smallest = average − (n−1) and largest = average + (n−1) for consecutive even/odd sets. Always re-read whether the smallest, middle or largest term is wanted.
Three consecutive natural numbers sum to 54. What is the smallest?
A16B17C18D19
The average of 4 consecutive even numbers is 17. What is the largest?
A16B18C20D22
The average of 7 consecutive odd numbers is 31. What is the largest?
