Sum of multiples or integers

7

1. Find the sum of all the multiples of 5 between 100 and 300 inclusive.
2. A geometric progression has a common ratio of \(- \frac { 2 } { 3 }\) and the sum of the first 3 terms is 35 . Find
   1. the first term of the progression,
   2. the sum to infinity.

1. (a) Calculate the sum of all the even numbers from 2 to 100 inclusive,

$$2 + 4 + 6 + \ldots \ldots + 100$$ (b) In the arithmetic series $$k + 2 k + 3 k + \ldots \ldots + 100$$ \(k\) is a positive integer and \(k\) is a factor of 100 .

1. Find, in terms of \(k\), an expression for the number of terms in this series.
2. Show that the sum of this series is $$50 + \frac { 5000 } { k }$$ (c) Find, in terms of \(k\), the 50th term of the arithmetic sequence $$( 2 k + 1 ) , ( 4 k + 4 ) , ( 6 k + 7 ) , \ldots \ldots ,$$ giving your answer in its simplest form.

1. (a) Find the sum of all the integers between 1 and 1000 which are divisible by 7 .
   (b) Hence, or otherwise, evaluate \(\sum _ { r = 1 } ^ { 142 } ( 7 r + 2 )\).
2. Solve the simultaneous equations

$$\begin{gathered} x - 3 y + 1 = 0 \\ x ^ { 2 } - 3 x y + y ^ { 2 } = 11 \end{gathered}$$

13. (a) Find the sum of all the integers between 1 and 1000 which are divisible by 7 .
(b) Hence, or otherwise, evaluate \(\sum _ { r = 1 } ^ { 142 } ( 7 r + 2 )\).
[0pt] [P1 June 2002 Question 1]