Special series: natural numbers, odd numbers, squares and cubes
Sums of the first n: natural numbers n(n + 1)/2, odd numbers n², squares n(n + 1)(2n + 1)/6, cubes [n(n + 1)/2]².
The special series
| Series | Sum | Example |
|---|---|---|
| 1 + 2 + 3 + ... + n | n(n + 1)/2 | 1 + 2 + ... + 50 = 50 × 51/2 = 1275 |
| 2 + 4 + 6 + ... + 2n (twice the natural numbers) | n(n + 1) | 2 + 4 + ... + 100 = 50 × 51 = 2550 |
| 1 + 3 + 5 + ... + (2n − 1) | n² | 1 + 3 + ... + 39 has n = 20 terms, so the sum is 400 |
| 1² + 2² + ... + n² | n(n + 1)(2n + 1)/6 | 1² + ... + 10² = 10 × 11 × 21/6 = 385 |
| 1³ + 2³ + ... + n³ | [n(n + 1)/2]² | 1³ + ... + 10³ = 55² = 3025 |
Spot the link
The sum of the first n cubes is the square of the sum of the first n natural numbers. So if 1 + 2 + ... + n = 55, then 1³ + 2³ + ... + n³ = 55² = 3025. For a sum that does not start at 1, subtract: (sum up to the last term) − (sum up to the term just before the first).
Example 1: a sum that does not start at 1
Find the value of 11² + 12² + 13² + ... + 20².
- Formula: 1² + 2² + ... + n² = n(n + 1)(2n + 1)/6.
- Write it as (1² + 2² + ... + 20²) − (1² + 2² + ... + 10²).
- 1² + ... + 20² = 20 × 21 × 41/6 = 2870
- 1² + ... + 10² = 10 × 11 × 21/6 = 385
- Difference: 2870 − 385 = 2485
11² + 12² + ... + 20² = 2485
Example 2: odd numbers and cubes
(i) Find 1 + 3 + 5 + ... + 75. (ii) If 1 + 2 + 3 + ... + n = 171, find 1³ + 2³ + ... + n³.
- (i) The last term is l = 75, so the number of terms is (75 + 1)/2 = 38.
- Sum of the first n odd numbers = n² = 38² = 1444.
- (ii) 1³ + 2³ + ... + n³ = [n(n + 1)/2]² = (1 + 2 + ... + n)².
- So the sum of cubes = 171² = 29241.
(i) 1444 (ii) 29241
Example 3: a stacking word problem
A fruit seller stacks oranges in a square pyramid. The top layer has 1 orange, the next 4, then 9, and so on. The bottom layer has 64 oranges. How many oranges are in the stack?
- The layers hold 1², 2², 3², ..., 8² oranges, since 64 = 8².
- Formula: 1² + 2² + ... + n² = n(n + 1)(2n + 1)/6.
- Total = 1² + 2² + ... + 8² = 8 × 9 × 17/6
- = 1224/6 = 204.
There are 204 oranges in the stack.
Fruit sellers at Koyambedu market often stack oranges in square pyramids, and the count is a sum of squares. A kolam with rows of 1, 3, 5, 7, ... dots uses n² dots for n rows.
2 marks: “Find the sum 1 + 2 + 3 + ... + 80.” n(n + 1)/2 = 80 × 81/2 = 3240.
2 marks: “Find the sum 1³ + 2³ + ... + 14³.” [n(n + 1)/2]² = (14 × 15/2)² = 105² = 11025.
5 marks: “Find the value of 16² + 17² + ... + 30².” (1² + ... + 30²) − (1² + ... + 15²) = 30 × 31 × 61/6 − 15 × 16 × 31/6 = 9455 − 1240 = 8215.
Cubes are the square of the natural-number sum: 1³ + ... + n³ = (1 + ... + n)².