Sigma Notation (AQA A-Level Mathematics): Revision Notes

The sigma notation is written as: $\sum_{k=m}^{n} a_k$

This represents the sum of the terms $\ a_k$ from $\ k = m \ to \ k = n .$ Here's what each part means:

• $\ \Sigma$ denotes summation.
• $\ k$ is the index of summation, which starts at the lower limit $\ m$ and increases to the upper limit $\ n .$
• $\ a_k$ is the general term of the sequence you're summing.

Example 1:

Sum the first $5$ natural numbers:

$\sum_{k=1}^{5} k = 1 + 2 + 3 + 4 + 5 = :highlight[15]$

Example 2:

Sum the squares of the first 4 natural numbers:

$\sum_{k=1}^{4} k^2 = 1^2 + 2^2 + 3^2 + 4^2 = 1 + 4 + 9 + 16 = :highlight[30]$

Step-by-Step Breakdown:

1. Identify the General Term $\ a_k :$ This is the expression that changes as $\ k$ changes. For example, in$\ \sum_{k=1}^{4} k^2$, the general term $\ a_k = k^2 .$

2. Determine the Limits of Summation: The lower limit $m$ is where $\ k$ starts, and the upper limit $\ n$ is where $\ k$ ends. In the example $\ \sum_{k=1}^{4} k^2 \, \ k$ starts at $1$ and ends at $4$.

3. Expand and Calculate: Substitute the values of $\ k \ from \ m \ to \ n$ into the general term and add them together.

Example Exam Question:

Question: Evaluate the sum:

$\sum_{k=1}^{6} (2k + 1)$

[3 marks]

Solution:

4. Expand the Summation: $\sum_{k=1}^{6} (2k + 1) = (2(1) + 1) + (2(2) + 1) + (2(3) + 1) + (2(4) + 1) + (2(5) + 1) + (2(6) + 1)$ $= 3 + 5 + 7 + 9 + 11 + 13$
5. Calculate the Sum: $3 + 5 + 7 + 9 + 11 + 13 = :highlight[48]$ Final Answer: $\sum_{k=1}^{6} (2k + 1) = :highlight[48]$