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# What is the value of 1 + 4 + 2 + 8 + 3 + 12 + 4 + 16 + ... + 24 + 96 + 25 + 100?

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What is the value of 1 + 4 + 2 + 8 + 3 + 12 + 4 + 16 + ... + 24 + 96 + 25 + 100?

May 20, 2023

#1
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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

May 20, 2023
#2
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[1 + 2 + 3 +4 +5 +...........+ 25]  +  [4 + 8 + 12 + 16 + 20 +..........+100]=

[25  x  26] / 2 ==325         +        [100 - 4==96 / 4 + 1==25 terms=

325                          +       [4 + 100] / 2  x  25     ==1,300

325                          +                  1,300                  ==1,625

May 21, 2023