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Calculate the sum of the given number of terms in each arithmetic sequence.
1. Find the sum of the first 25 terms of the sequence defined by $a_n = 5n - 7$.
2. Determine the sum of the first 20 terms of the sequence defined by $a_j = \frac{3}{4} + \frac{1}{4}j$.
42
1
Suppose $t_n = 3n + 2$, for $n$ being an integer.
(i) Use summation notation to express $t_1 + t_2 + t_3 + \cdots + t_{50}$.
(ii) Calculate the sum you wrote in part (i).
187
54
Look at the arithmetic series $7 + 12 + 17 + 22 + \cdots + 92$:
(a) Derive the formula for the general term $t_m$ in the form $bm + c$.
(b) Using this, write the series sum using sigma notation.
295
54
Look at the sum $T = \sum_{m=5}^q (3m - 2)$, where $q$ is a positive integer greater than 5:
(a) Identify the first three terms of the series.
(b) Determine the total number of terms in the series.
72
52
