What is the value of 1 + 4 + 2 + 8 + 3 + 12 + 4 + 16 + ... + 24 + 96 + 25 + 100?
The given expression can be rewritten as follows:
1 + 4 + 2 + 8 + 3 + 12 + 4 + 16 + ... + 24 + 96 + 25 + 100 = (1 + 2 + 3 + 4 + ... + 25) + (4 + 8 + 12 + ... + 96 + 100)
The first sum is an arithmetic series with first term 1, last term 25, and common difference 1. The sum of an arithmetic series is given by the following formula:
S_n = \frac{n(a_1 + a_n)}{2}
where n is the number of terms, a1 is the first term, and an is the last term. In this case, n=25, a1=1, and an=25, so the sum of the first series is:
S_1 = \frac{25(1 + 25)}{2} = 312.5
The second sum is an arithmetic series with first term 4, last term 100, and common difference 4. The sum of an arithmetic series is given by the following formula:
S_n = \frac{n(a_1 + a_n)}{2}
where n is the number of terms, a1 is the first term, and an is the last term. In this case, n=25, a1=4, and an=100, so the sum of the second series is:
S_2 = \frac{25(4 + 100)}{2} = 562.5
Therefore, the value of the given expression is:
1 + 4 + 2 + 8 + 3 + 12 + 4 + 16 + ... + 24 + 96 + 25 + 100 = S_1 + S_2 = 312.5 + 562.5 = 875
