Sum of multiples or integers

Find the sum of all multiples of k, or all integers in a range satisfying a condition, by recognizing as an arithmetic series.

5 questions · Moderate -0.5

1.04h Arithmetic sequences: nth term and sum formulae
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Edexcel C1 2011 June Q9
9 marks Moderate -0.8
  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.
CAIE P1 2010 June Q7
8 marks Moderate -0.8
  1. Find the sum of all the multiples of 5 between 100 and 300 inclusive. [3]
  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, [3]
    2. the sum to infinity. [2]
Edexcel C1 Q13
6 marks Moderate -0.3
  1. Find the sum of all the integers between 1 and 1000 which are divisible by 7. [3]
  2. Hence, or otherwise, evaluate \(\sum_{r=1}^{142} (7r + 2)\). [3]
Edexcel C1 Q1
6 marks Moderate -0.3
  1. Find the sum of all the integers between 1 and 1000 which are divisible by 7. [3]
  2. Hence, or otherwise, evaluate \(\sum_{r=1}^{142}(7r + 2)\). [3]
Edexcel C1 Q8
9 marks Moderate -0.3
  1. Prove that the sum of the first \(n\) positive integers is given by $$\frac{1}{2}n(n + 1).$$ [4]
  2. Hence, find the sum of
    1. the integers from 100 to 200 inclusive,
    2. the integers between 300 to 600 inclusive which are divisible by 3.
    [5]