KA-0058Clever regrouping (pair to round numbers)
4 points, difficulty 2 of 3Level 5–6about 90sWhat is 2 + 4 + 6 + ... + 100?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Pair the outside in, then count the pairs.
2The strategy
Every term is even, so the whole sum is twice the sum of 1 to 50. Alternatively pair 2 with 100, 4 with 98, and so on, and count how many pairs fifty terms give.
3The full solution
There are 50 terms, so 25 pairs, each adding to 102. That is 25 x 102 = 2550.
Solution
The reliable way
There are 50 even numbers from 2 to 100. Pair the outside in: 2 + 100 = 102, 4 + 98 = 102, and so on, giving 25 pairs of 102. The total is 25 x 102 = 2550. Check the other way: the sum is twice 1 + 2 + ... + 50 = 2 x 1275 = 2550. The transferable idea: two routes that agree is worth more than one route checked twice.
The elegant way
Every term is double the matching term of 1 to 50, so the sum is twice 1275, which is 2550.
Why this is on the test: Counting the pairs is where this goes wrong: fifty terms make twenty-five pairs, not fifty and not forty-nine.