Q1.
Later in this series, we get the following set of consecutive numbers: 91 + 93 + 95 + 97 + 99 + 101 + 103 + 105 + 107 + 109. Can you tell what this sum is without doing the calculation?
Answer
1000, which is 103.
Count the terms: 91, 93, …, 109 → 10 consecutive odd numbers
In the pattern, a run of n consecutive odd numbers adds to n3
So the sum = 103 = 1000
In the pattern, a run of n consecutive odd numbers adds to n3
So the sum = 103 = 1000
Why it works: The rows use 1, then 2, then 3, … consecutive odd numbers, giving 13, 23, 33, …. A run of 10 must therefore give 103. You can also see it from the middle: the terms pair off around the centre — 91 + 109 = 200, 93 + 107 = 200, and so on, five pairs of 200, giving 5 × 200 = 1000.
Tip: Each row starts where the previous one stopped, so the runs never overlap. That is why adding all the rows up to n gives 13 + 23 + … + n3 as a single unbroken block of odd numbers.