Write it twice, back to front
Add the list to its own reverse. Every column gives the same total.
Gauss and the 100 numbers
Add 1 + 2 + 3 + … + 100. Gauss did it in seconds when he was 10.
He wrote the sum, and then wrote it again backwards underneath: 1, 2, 3, …, 99, 100 and then 100, 99, …, 2, 1. Then he added the two rows column by column.
What do you think he saw about the totals of the columns?
What this lesson covers
The idea
The sum of the first n terms of an AP is S = (n/2)[2a + (n – 1)d], obtained by adding the sum to itself written in reverse order; knowing any three of S, a, d, n gives the fourth.
Gauss and the 100 numbers
Add 1 + 2 + 3 + … + 100. Gauss did it in seconds when he was 10.
He wrote the sum, and then wrote it again backwards underneath: 1, 2, 3, …, 99, 100 and then 100, 99, …, 2, 1. Then he added the two rows column by column.
What do you think he saw about the totals of the columns?
- Every column adds to the same total
- Every column adds to a different total
- The column totals get bigger and bigger
Add the row to its reverse
Row 1 is an AP. Row 2 is the same AP written backwards. Tap each column to add its two numbers. You can change the number of terms n.
The sum formula
Write S = a + (a + d) + … + [a + (n − 1)d], and again in reverse. Every column adds to 2a + (n − 1)d, and there are n columns. So 2S = n[2a + (n − 1)d], and S = (n/2)[2a + (n − 1)d].
Why is every column the same? Going to the right, the top number goes up by d and the bottom number goes down by d, so the total cannot change.
The formula joins four numbers: S, a, d and n. If you know any three of them, you can find the fourth.
Notes
S = (n/2)[2a + (n − 1)d]. Write the sum twice, one reversed, and add: n columns, each 2a + (n − 1)d, give 2S. Knowing any three of S, a, d, n gives the fourth.
Check yourself
Find the sum of the first 22 terms of the AP 8, 3, −2, …
Answer: -979
S = (22/2)[2 × 8 + 21 × (−5)] = 11 × (16 − 105) = 11 × (−89) = −979.
Shakila puts ₹100 in a money box on her daughter's first birthday, ₹150 on the second, ₹200 on the third, and so on. How many rupees are in the box after the 21st birthday?
Answer: 12600
S = (21/2)[2 × 100 + 20 × 50] = (21/2) × 1200 = 21 × 600 = 12600.
The sum of the first 14 terms of an AP is 1050, and its first term is 10. Find d.
Answer: 10
1050 = 7(20 + 13d), so 20 + 13d = 150, so 13d = 130 and d = 10.
The sum formula has four numbers S, a, d and n. If S, n and a are known, can you find d?
- Yes. It is one equation with only d unknown. — correct. Yes! Any three of S, a, d and n give the fourth.
- No. You also need the last term *l*.. *l* is built from a, d and n. The formula already has all you need.
- No. You can only find S from a, d and n.. The formula is an equation. You can solve it for any one of its letters if the other three are known.