Sum of specific range of terms

7 The first term of an arithmetic progression is 1.5 and the sum of the first ten terms is 127.5 .

1. Find the common difference.
2. Find the sum of all the terms of the arithmetic progression whose values are between 25 and 100 .

7 In an arithmetic progression the first term is 5 and the common difference is 3 . The \(n\)th term of the progression is denoted by \(u _ { n }\).

1. Find the value of \(u _ { 20 }\).
2. Show that \(\sum _ { n = 10 } ^ { 20 } u _ { n } = 517\).
3. Find the value of \(N\) such that \(\sum _ { n = N } ^ { 2 N } u _ { n } = 2750\).

3 An arithmetic series has fifth term 46 and twentieth term 181.

1. 1. Show that the common difference is 9 .
   2. Find the first term.
2. Find the sum of the first 20 terms of the series.
3. The \(n\)th term of the series is \(u _ { n }\). Given that the sum of the first 50 terms of the series is 11525 , find the value of $$\sum _ { n = 21 } ^ { 50 } u _ { n }$$

2 The arithmetic series $$51 + 58 + 65 + 72 + \ldots + 1444$$ has 200 terms.

1. Write down the common difference of the series.
2. Find the 101st term of the series.
3. Find the sum of the last 100 terms of the series.

4 An arithmetic series has first term \(a\) and common difference \(d\).
The sum of the first 21 terms is 168 .

1. Show that \(a + 10 d = 8\).
2. The sum of the second term and the third term is 50 . The \(n\)th term of the series is \(u _ { n }\).
   1. Find the value of \(u _ { 12 }\).
   2. Find the value of \(\sum _ { n = 4 } ^ { 21 } u _ { n }\).

Evaluate $$\sum_{r=10}^{30} (7 + 2r).$$ [4]

A sequence is given by the following. $$u_1 = 3$$ $$u_{n+1} = u_n + 5$$

1. Write down the first 4 terms of this sequence. [1]
2. Find the sum of the 51st to the 100th terms, inclusive, of the sequence. [4]

The third term of an arithmetic progression is 24. The tenth term is 3. Find the first term and the common difference. Find also the sum of the 21st to 50th terms inclusive. [5] Simplify

An arithmetic progression has first term 7 and third term 12.

1. Find the 20th term of this progression. [2]
2. Find the sum of the 21st to the 50th terms inclusive of this progression. [3]

Showing all your working, evaluate

1. \(\sum_{r=3}^{50} (4r + 5)\) [4]
2. \(\sum_{r=2}^{\infty} \left(540 \times \left(\frac{1}{3}\right)^r\right)\). [3]