Answer :
To find the value of the series, we need to identify and sum all the terms in the given series:
[tex]\[ 1, 4, 2, 8, 3, 12, 4, 16, 5, 20, 6, 24, 7, 28, 8, 32, 9, 36, 10, 40, 11, 44, 12, 48, 13, 52, 14, 56, 15, 60, 16, 64, 17, 68, 18, 72, 19, 76, 20, 80, 21, 84, 22, 88, 23, 92, 24, 96, 25, 100. \][/tex]
### Step-by-Step Summation:
1. Identify the given numbers in the series.
2. Sum all these numbers sequentially.
Let's list and sum them step-by-step:
[tex]\[ \begin{align*} 1 + 4 &= 5, \\ 5 + 2 &= 7, \\ 7 + 8 &= 15, \\ 15 + 3 &= 18, \\ 18 + 12 &= 30, \\ 30 + 4 &= 34, \\ 34 + 16 &= 50, \\ 50 + 5 &= 55, \\ 55 + 20 &= 75, \\ 75 + 6 &= 81, \\ 81 + 24 &= 105, \\ 105 + 7 &= 112, \\ 112 + 28 &= 140, \\ 140 + 8 &= 148, \\ 148 + 32 &= 180, \\ 180 + 9 &= 189, \\ 189 + 36 &= 225, \\ 225 + 10 &= 235, \\ 235 + 40 &= 275, \\ 275 + 11 &= 286, \\ 286 + 44 &= 330, \\ 330 + 12 &= 342, \\ 342 + 48 &= 390, \\ 390 + 13 &= 403, \\ 403 + 52 &= 455, \\ 455 + 14 &= 469, \\ 469 + 56 &= 525, \\ 525 + 15 &= 540, \\ 540 + 60 &= 600, \\ 600 + 16 &= 616, \\ 616 + 64 &= 680, \\ 680 + 17 &= 697, \\ 697 + 68 &= 765, \\ 765 + 18 &= 783, \\ 783 + 72 &= 855, \\ 855 + 19 &= 874, \\ 874 + 76 &= 950, \\ 950 + 20 &= 970, \\ 970 + 80 &= 1050, \\ 1050 + 21 &= 1071, \\ 1071 + 84 &= 1155, \\ 1155 + 22 &= 1177, \\ 1177 + 88 &= 1265, \\ 1265 + 23 &= 1288, \\ 1288 + 92 &= 1380, \\ 1380 + 24 &= 1404, \\ 1404 + 96 &= 1500, \\ 1500 + 25 &= 1525, \\ 1525 + 100 &= 1625. \end{align*} \][/tex]
### Conclusion:
After summing all the terms of the series, we find that the value of the series is
[tex]\[ 1625. \][/tex]
[tex]\[ 1, 4, 2, 8, 3, 12, 4, 16, 5, 20, 6, 24, 7, 28, 8, 32, 9, 36, 10, 40, 11, 44, 12, 48, 13, 52, 14, 56, 15, 60, 16, 64, 17, 68, 18, 72, 19, 76, 20, 80, 21, 84, 22, 88, 23, 92, 24, 96, 25, 100. \][/tex]
### Step-by-Step Summation:
1. Identify the given numbers in the series.
2. Sum all these numbers sequentially.
Let's list and sum them step-by-step:
[tex]\[ \begin{align*} 1 + 4 &= 5, \\ 5 + 2 &= 7, \\ 7 + 8 &= 15, \\ 15 + 3 &= 18, \\ 18 + 12 &= 30, \\ 30 + 4 &= 34, \\ 34 + 16 &= 50, \\ 50 + 5 &= 55, \\ 55 + 20 &= 75, \\ 75 + 6 &= 81, \\ 81 + 24 &= 105, \\ 105 + 7 &= 112, \\ 112 + 28 &= 140, \\ 140 + 8 &= 148, \\ 148 + 32 &= 180, \\ 180 + 9 &= 189, \\ 189 + 36 &= 225, \\ 225 + 10 &= 235, \\ 235 + 40 &= 275, \\ 275 + 11 &= 286, \\ 286 + 44 &= 330, \\ 330 + 12 &= 342, \\ 342 + 48 &= 390, \\ 390 + 13 &= 403, \\ 403 + 52 &= 455, \\ 455 + 14 &= 469, \\ 469 + 56 &= 525, \\ 525 + 15 &= 540, \\ 540 + 60 &= 600, \\ 600 + 16 &= 616, \\ 616 + 64 &= 680, \\ 680 + 17 &= 697, \\ 697 + 68 &= 765, \\ 765 + 18 &= 783, \\ 783 + 72 &= 855, \\ 855 + 19 &= 874, \\ 874 + 76 &= 950, \\ 950 + 20 &= 970, \\ 970 + 80 &= 1050, \\ 1050 + 21 &= 1071, \\ 1071 + 84 &= 1155, \\ 1155 + 22 &= 1177, \\ 1177 + 88 &= 1265, \\ 1265 + 23 &= 1288, \\ 1288 + 92 &= 1380, \\ 1380 + 24 &= 1404, \\ 1404 + 96 &= 1500, \\ 1500 + 25 &= 1525, \\ 1525 + 100 &= 1625. \end{align*} \][/tex]
### Conclusion:
After summing all the terms of the series, we find that the value of the series is
[tex]\[ 1625. \][/tex]
