Series

A sequence lists terms. A series adds them.

From sequence to sum

If a sequence is $t_1,t_2,t_3,\ldots$, its first $n$ terms form the finite series

The symbol $S_n$ names a total, while $t_n$ names one term. Keeping them distinct is essential in financial, pattern, and modelling questions.

Sigma notation

The sum

can be written

The lower value tells where the index begins, the upper value where it ends, and the expression gives each term. There are $5-1+1=5$ terms.

Arithmetic series formula

For an arithmetic sequence with first term $a$, last term $t_n$, and $n$ terms,

Since $t_n=a+(n-1)d$, an equivalent formula is

Use whichever form matches the available information.

Why the arithmetic formula works

Write an arithmetic series forward and backward. Each vertical pair has the same sum $a+t_n$, and there are $n$ pairs across the two copies. Therefore $2S_n=n(a+t_n)$, giving the formula after division by $2$.

This pairing argument explains the structure rather than asking us to memorize it blindly.

Worked example: arithmetic total

Finite geometric series formula

For a geometric sequence with first term $a$ and common ratio $r\ne1$,

An equivalent form is $a(r^n-1)/(r-1)$. Choose one form and substitute signs carefully. If $r=1$, every term equals $a$, so $S_n=na$.

Why the geometric formula works

Start with

Multiply by $r$ and subtract the original sum. Most terms cancel:

Then $(r-1)S_n=a(r^n-1)$, which rearranges to the finite formula.

Worked example: geometric total

A ball drops $10$ m, then rebounds to $60\%$ of each previous drop height. The sum of the first five downward distances is

Using $a=10$, $r=0.6$, and $n=5$:

If total travel were requested, upward distances would also need to be included; modelling the wording matters.

Infinite geometric series preview

When $|r|<1$, terms approach zero quickly enough that an infinite geometric sum converges:

If $|r|\ge1$, the terms do not shrink appropriately and the infinite series does not have a finite sum. A finite series, however, can always be added term by term.

Choosing the correct structure

First decide whether the underlying sequence has a constant difference or constant ratio. Then identify whether the question asks for one term or a total. A phrase such as “in the 20th row” asks for $t_{20}$; “in the first 20 rows” asks for $S_{20}$.

Units provide another check: one payment is a term; cumulative money is a sum.

Common mistakes

Using a term formula for a total. Identify $t_n$ versus $S_n$.

Using the wrong number of terms. From index $p$ to $q$ inclusive there are $q-p+1$ terms.

Applying the arithmetic formula to a geometric pattern. Test difference and ratio first.

Dropping parentheses around $1-r^n$. Evaluate the exponent before subtraction.

Using $S_\infty$ when $|r|\ge1$. The convergence condition is required.

Quick self-check

• Is the question asking for a term or a sum?
• Is the sequence arithmetic or geometric?
• What are $a$, $d$ or $r$, and $n$?
• Does the formula match the information provided?
• Are the number of terms and units correct?
• Is an infinite sum allowed by $|r|<1$?

Related topics

• Arithmetic Sequences
• Geometric Sequences
• Exponential Functions
• Sigma Notation