Speedy Summations

In the video below, Alison works out \(\sum_{i=1}^{10} i\).

How could you adapt her method to work out the following sums?

• $\sum_{i=1}^{100} i$
   
• $2+4+6+\dots+96+98+100$
   
• $\sum_{k=1}^{20} (4k+12)$
   
• $37+42+47+52+\dots+102+107+112$
   
• The sum of the first $n$ terms of the sequence $a, (a+d), (a + 2d), (a + 3d) \dots$

After how many terms would $17+21+25+\dots$ be greater than $1000$?

Can you find the sum of all the integers less than $1000$ which are not divisible by $2$ or $3$?

Can you find a set of consecutive positive integers whose sum is 32?