Depending on the total of terms in the addition, it should make a square number. The formula to work this out is n², with n being the number of digits.
1. 'What is the sum of the first 30 odd numbers?'
You solve this by using the formula, so 30², which is 30 × 30, is 900.
Using the same method for question 2. 'What is the sum of the first 60 odd numbers?', you do 60², which is 60 × 60, is 3600.
'Can you make 3249 by adding odd numbers in this way?'
As any square number is the sum of consecutive odd numbers, and 3249 is a square number (it is the product of 57 × 57), it can be made in this way.
'1 + 3 +. . .+ 153'
As there are 77 terms, we use 77², which is 77×77, is 5929. We use the same method for the next.
'51 + 53 +. . .+ 153'
We already know what 1 + 3 +. . .+ 153, which is 5929, we use 5929 - the number of terms in 1 + 3 +. . .+ 49, which is 25, squared.
So 5929 - 25²
= 5929 - 625
= 5300
'2 + 4 + 6 +. . .+ 152 + 154'
In this sum, it is (2 + 154) + (4 + 152) +. . .+ (76 + 80) + 78. There are 77 terms, so there are 38.5 pairs of 2 + 154, which is 156, in total, so you use 38.5 × 156, which is 6006. You can do a similar calculation for the last question.