Write the sum in sigma notation: _r=1^10r Quote the standard result for r: _r=1^Nr=(1)/(2)N(N+1) State the final answer: _r=1^10r=55 Answer: 55

Write the sum in sigma notation: _r=1^12r^2 Quote the standard result for r^2: _r=1^Nr^2=(1)/(6)N(N+1)(2N+1) State the final answer: _r=1^12r^2=650 Answer: 650

Write the sum in sigma notation: _r=1^6r^3 Quote the standard result for r^3: _r=1^Nr^3=(1)/(4)N^2(N+1)^2 Simplify to the final form: 441 State the final answer: _r=1^6r^3=441 Answer: 441

Write the sum in sigma notation: _r=1^20r Quote the standard result for r: _r=1^Nr=(1)/(2)N(N+1) State the final answer: _r=1^20r=210 Answer: 210

Write the sum in sigma notation: _r=1^8r^2 Quote the standard result for r^2: _r=1^Nr^2=(1)/(6)N(N+1)(2N+1) State the final answer: _r=1^8r^2=204 Answer: 204

Series Worked Solutions — Further Maths Maths

Fully worked, step-by-step solutions to Further Maths Series questions. See exactly how to solve problems on standard-results, sum-of-integers, sum-of-squares, sum-of-cubes.

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Further Maths70 questionsStep-by-step solutions

Question 1

2 markseasy

Evaluate

\sum_{r=1}^{10}r

Worked solution

Write the sum in sigma notation
$\sum_{r=1}^{10}r$
Identify the summand and the limits of the sum.
Quote the standard result for $\sum r$
$\sum_{r=1}^{N}r=\frac{1}{2}N\left(N+1\right)$
The sum of the first $N$ integers.
State the final answer
$\sum_{r=1}^{10}r=55$
This is the value of the sum.

Answer

55

Question 2

2 markseasy

Evaluate

\sum_{r=1}^{12}r^{2}

Worked solution

Write the sum in sigma notation
$\sum_{r=1}^{12}r^{2}$
Identify the summand and the limits of the sum.
Quote the standard result for $\sum r^{2}$
$\sum_{r=1}^{N}r^{2}=\frac{1}{6}N\left(N+1\right)\left(2N+1\right)$
The sum of the first $N$ squares.
State the final answer
$\sum_{r=1}^{12}r^{2}=650$
This is the value of the sum.

Answer

650

Question 3

2 markseasy

Evaluate

\sum_{r=1}^{6}r^{3}

Worked solution

Write the sum in sigma notation
$\sum_{r=1}^{6}r^{3}$
Identify the summand and the limits of the sum.
Quote the standard result for $\sum r^{3}$
$\sum_{r=1}^{N}r^{3}=\frac{1}{4}N^{2}\left(N+1\right)^{2}$
The sum of the first $N$ cubes.
Simplify to the final form
$441$
This is the fully simplified answer.
State the final answer
$\sum_{r=1}^{6}r^{3}=441$
This is the value of the sum.

Answer

441

Question 4

2 markseasy

Evaluate

\sum_{r=1}^{20}r

Worked solution

Write the sum in sigma notation
$\sum_{r=1}^{20}r$
Identify the summand and the limits of the sum.
Quote the standard result for $\sum r$
$\sum_{r=1}^{N}r=\frac{1}{2}N\left(N+1\right)$
The sum of the first $N$ integers.
State the final answer
$\sum_{r=1}^{20}r=210$
This is the value of the sum.

Answer

210

Question 5

2 markseasy

Evaluate

\sum_{r=1}^{8}r^{2}

Worked solution

Write the sum in sigma notation
$\sum_{r=1}^{8}r^{2}$
Identify the summand and the limits of the sum.
Quote the standard result for $\sum r^{2}$
$\sum_{r=1}^{N}r^{2}=\frac{1}{6}N\left(N+1\right)\left(2N+1\right)$
The sum of the first $N$ squares.
State the final answer
$\sum_{r=1}^{8}r^{2}=204$
This is the value of the sum.

Answer

204

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