Finding Coefficients from a Sum in Sigma Notation

Sigma notation is a useful way to express the sum of a large number of terms. When we want to find particular terms or coefficients, we don’t always have to expand the whole expression to find it.

Example 1: Write the expression 3+6+9+12+…+60 in sigma notation.
• notice that we are adding multiples of 3;
• so we can write this sum as ∑20n=1 3n.

We can also use sigma notation when we have variables in our terms.
Example 2: Write the expression 3x+6x2+9x3+12x4+⋯+60x20 in sigma notation.
• note from Example 1 the numbers are multiples of 3 and can be represented by 3n where n: 1, 2, …, 20;
• we also have powers of x which increase by 1 in each subsequent term;
• so we can write this sum as ∑20n=1 3nxn.
The numbers in front of the variables are called coefficients. In Example 2 the coefficient of x is 3 and the coefficient of x2 is 6.

Example 3: Find the coefficient of x4 in ∑8k=0 (4k+3) xk.
• the terms in this sum look like (4k+3) xk;
• the terms with x4 occurs when k=4 i.e.(4⋅4+3) x4=19x4;
• the coefficient of x4 is 19.

Example 4: Find the coefficient of x7 in ∑8k=0 (4k+3) x(k+2).
• a typical term is of the form (4k+3) x(k+2);
• a the term with x7 occurs when k+2=7 i.e. k=5;
• we have (4⋅5+3) x(5+2)=23x7;
• a the Coefficient of x7 is 23.

Example 5: Find the Coefficient of x2 in (3+x)∑8k=0 (4k+3) xk
• we can think of this as 3∑8k=0 (4k+3) xk+x8k=0 (4k+3) xk;
• the term with x2 can be obtained by taking k=2 from the first part of this expression to get 3(4⋅2+3)x2=33x2 and then taking k=1 from the second part of this expression to get x(4⋅1+3)x1=7x2;
• combining these we get 33x2+7x2=40x2;
• so the coefficient of x2 is 40.