Edexcel IAL - Further Pure Mathematics 1- 7.1 Summation of Simple Finite Series- Study notes - New syllabus
Edexcel IAL – Further Pure Mathematics 1- 7.1 Summation of Simple Finite Series -Study notes- New syllabus
Edexcel IAL – Further Pure Mathematics 1- 7.1 Summation of Simple Finite Series -Study notes -Edexcel A level Maths- per latest Syllabus.
Key Concepts:
- 7.1 Summation of Simple Finite Series
Summation of Simple Finite Series
In many problems we need to add a sequence of numbers written in compact form using the sigma notation \( \sum \). A finite series means we are adding a fixed number of terms.
For example:
\( \sum_{r=1}^{n} r = 1 + 2 + 3 + \dots + n \)
Standard Summation Formulae
| Sum | Formula |
| \( \sum_{r=1}^{n} r \) | \( \dfrac{n(n+1)}{2} \) |
| \( \sum_{r=1}^{n} r^2 \) | \( \dfrac{n(n+1)(2n+1)}{6} \) |
| \( \sum_{r=1}^{n} r^3 \) | \( \left(\dfrac{n(n+1)}{2}\right)^2 \) |
These formulae allow us to evaluate sums quickly without listing all terms.
Using Linearity of Summation
Summation is linear, which means:
\( \sum (a_r + b_r) = \sum a_r + \sum b_r \)
\( \sum k a_r = k \sum a_r \)
This allows complicated sums to be broken into simpler parts.
Example: \( \sum_{r=1}^{n} r(r^2+2) \)
First expand:
\( r(r^2+2) = r^3 + 2r \)
So:
\( \sum_{r=1}^{n} r(r^2+2) = \sum_{r=1}^{n} r^3 + 2\sum_{r=1}^{n} r \)
Now use the standard formulae.
Example
Find \( \sum_{r=1}^{10} r \).
▶️ Answer / Explanation
\( \sum_{r=1}^{10} r = \dfrac{10 \times 11}{2} = 55 \)
Example
Find \( \sum_{r=1}^{5} r^2 \).
▶️ Answer / Explanation
\( \sum_{r=1}^{5} r^2 = \dfrac{5 \times 6 \times 11}{6} = 55 \)
Example
Find \( \sum_{r=1}^{n} r(r^2 + 2) \).
▶️ Answer / Explanation
\( \sum_{r=1}^{n} (r^3 + 2r) = \sum_{r=1}^{n} r^3 + 2\sum_{r=1}^{n} r \)
\( = \left(\dfrac{n(n+1)}{2}\right)^2 + 2 \cdot \dfrac{n(n+1)}{2} \)
\( = \dfrac{n^2(n+1)^2}{4} + n(n+1) \)
\( = \dfrac{n(n+1)\big(n(n+1)+4\big)}{4} \)