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 ∑^{20}_{n=1} 3*n*.

We can also use sigma notation when we have variables in our terms.

Example 2: Write the expression 3*x*+6*x*^{2}+9*x*^{3}+12*x*^{4}+⋯+60*x*^{20} in sigma notation.

• note from Example 1 the numbers are multiples of 3 and can be represented by 3*n* 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 ∑^{20}_{n=1} 3*n**x ^{n}*.

The numbers in front of the variables are called coefficients. In Example 2 the coefficient of

*x*is 3 and the coefficient of

*x*

^{2}is 6.

Example 3: Find the coefficient of *x*^{4} in ∑^{8}_{k=0} (4*k*+3) *x ^{k}*.

• the terms in this sum look like (4

*k*+3)

*x*;

^{k}• the terms with

*x*

^{4}occurs when

*k*=4 i.e.(4⋅4+3)

*x*

^{4}=19

*x*

^{4};

• the coefficient of

*x*

^{4}is 19.

Example 4: Find the coefficient of *x*^{7} in ∑^{8}_{k=0} (4*k*+3) *x*^{(k+2)}.

• a typical term is of the form (4*k*+3) *x*^{(k+2)};

• a the term with *x*^{7} occurs when *k*+2=7 i.e. *k*=5;

• we have (4⋅5+3) *x*^{(5+2)}=23*x*^{7};

• a the Coefficient of *x*^{7} is 23.

Example 5: Find the Coefficient of *x*^{2} in (3+*x*)∑^{8}_{k=0} (4*k*+3) *x ^{k}*

• we can think of this as 3∑

^{8}

_{k=0}(4

*k*+3)

*x*+

^{k}*x*∑

^{8}

_{k=0}(4

*k*+3)

*x*;

^{k}• the term with

*x*

^{2}can be obtained by taking

*k*=2 from the first part of this expression to get 3(4⋅2+3)

*x*

^{2}=33

*x*

^{2}and then taking

*k*=1 from the second part of this expression to get x(4⋅1+3)

*x*

^{1}=7

*x*

^{2};

• combining these we get 33

*x*

^{2}+7

*x*

^{2}=40

*x*

^{2};

• so the coefficient of

*x*

^{2}is 40.