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]
