Sigma notation and arithmetic seriesAQA A-Level Maths: Subtopic test

1. 1
   An arithmetic sequence has first term $7$ and common difference $4$.

   (a)

   Find the $20$th term.

   [1 mark]

   • A$79$
   • B$87$
   • C$83$
   • D$80$

   (b)

   Find the sum of the first $20$ terms.

   [1 mark]

   • A$900$
   • B$940$
   • C$1800$
   • D$817$

   (c)

   Find the least value of $n$ for which the sum of the first $n$ terms exceeds $5000$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A series is given in sigma notation as $\sum_{r=1}^{n}(3r+2)$.

   (a)

   Find the value of the series when $n=4$.

   [1 mark]

   • A$30$
   • B$32$
   • C$14$
   • D$38$

   (b)

   Which expression gives the sum to $n$ terms, $\sum_{r=1}^{n}(3r+2)$?

   [1 mark]

   • A$\frac{n(3n+2)}{2}$
   • B$\frac{n(3n+7)}{2}$
   • C$\frac{n(3n+5)}{2}$
   • D$\frac{3n+7}{2}$

   (c)

   Find $\sum_{r=10}^{30}(3r+2)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   An arithmetic series has third term $17$ and tenth term $45$.

   (a)

   Find the first term and the common difference.

   [3 marks]

   (b)

   The sum of the first $n$ terms is $1425$. Find $n$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   An arithmetic series has first term $a$ and common difference $d$, where $a$ and $d$ are positive integers.

   (a)

   The sum of the first $8$ terms is $148$ and the sum of the first $12$ terms is $342$. Find $a$ and $d$.

   [6 marks]

   (b)

   Using your values from (a):
   (i) show that the $r$th term is $5r-4$;
   (ii) show that $\sum_{r=1}^{n}(5r-4)=\frac{n(5n-3)}{2}$;
   (iii) find $\sum_{r=11}^{20}(5r-4)$.

   [6 marks]

   Total for question 4: 12 marks

End of questions