To solve the first one, let's try to first find the 26th number. Notice there is a pattern:

To get to the second number from the first, we take one step and go forward 4.

To get to the third number from the first, we take 2 steps and go forward 8.

To get to the fourth pattern from the first, we take 3 steps and go forward 12.

See the pattern? To get to the nth term from the first, we take n - 1 steps and go forward 4(n - 1). (It is multiplied by 4 because the difference between two neighbouring numbers is 4.) We want to get to the 26th term, so we take 25 steps and go forward 4 * 25 = 100. So, the 26th term is 100 + 7 = 107.

To find the sum of 7 + 11 + 13 + 17 + 21 + ...... + 99 + 103 + 107, imagine pairing the 1st with the 26th, the 2nd with the 25th, the 3rd with the 24th, and so on. There will be 26 / 2 = 13 pairs with each one summing to 7 + 107 = 114. So, the sum is 13 * 114 = 1482.

To solve the second one, first note that the weird looking symbol just means summation. The n = 1 on the bottom tells us what number to plug into the expression on the side, 4n + 1. Then, we repeat with 2, 3, 4, 5, all the way up to the number on the top, 32. Then, we sum all of these values. So, what we want is 5 + 9 + 13 + 17 + ..... + 121 + 125 + 129. Pair them up again to get 32 / 2 = 16 pairs with each one having a sum of 5 + 129 = 134. Multiply to get 16 * 134 = 1600 + 480 + 64 = 2080 + 64 = 2114.

To solve the third one, we already know what we want: 1 + 2 + 3 + 4 + 5 + ..... + 199 + 200. Pair these up to get 200 / 2 = 100 pairs, with each one summing to 1 + 200 = 201. So, the sum is 20100.

The answers are:

1. 1482

2. 2114

3. 20100

Hope this helped,

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