Arithmetic Sequences

An arithmetic sequence (or arithmetic progression) is a number pattern in which each term is obtained by adding a fixed number to the previous term. This fixed number is called the common difference.

General Form of an Arithmetic Sequence:

\( a,\ a + d,\ a + 2d,\ a + 3d,\ \ldots \)

Where:

• \( a \) = first term
• \( d \) = common difference
• \( n \) = position (term number)

Formula for the $n^{th}$ Term

The general formula to find the \(n^{th}\) term of an arithmetic sequence is:

\( T_n = a + (n – 1)d \)

Formula for the Sum of First $n$ Terms

The sum of the first \(n\) terms of an arithmetic sequence is given by:

\( S_n = \frac{n}{2} \left[2a + (n – 1)d\right] \)

or

\( S_n = \frac{n}{2}(a + l) \)

Where:

• \( l \) is the last term (can be calculated using \( l = a + (n – 1)d \))

Important Properties of Arithmetic Sequences

• The difference between any two consecutive terms is constant.
• The graph of an arithmetic sequence (with \(n\) on x-axis, term on y-axis) is a straight line.
• If the common difference is:
  • Positive → sequence increases.
  • Negative → sequence decreases.
  • Zero → sequence is constant.

Geometric Sequences

A geometric sequence is a sequence where each term is obtained by multiplying the previous term by a fixed number called the common ratio.

This is different from an arithmetic sequence, where a fixed number is added.

General Form of a Geometric Sequence:

\( a,\ ar,\ ar^2,\ ar^3,\ ar^4,\ \ldots \)

Where:

• \( a \) is the first term
• \( r \) is the common ratio (found by dividing any term by the previous one)

Formula for the $n^{th}$ Term:

The \( n^{th} \) term of a geometric sequence is given by:

\( u_n = ar^{n-1} \)

To find the common ratio \( r \):

\( r = \frac{\text{any term}}{\text{previous term}} \)

Important Notes:

• If \( r > 1 \), the sequence is increasing.
• If \( 0 < r < 1 \), the sequence is decreasing.
• If \( r = 1 \), the sequence is constant.
• If \( r < 0 \), the sequence alternates in sign.

To find a missing term or the position of a term, use:

\( u_n = ar^{n-1} \Rightarrow n = 1 + \log_r \left(\frac{u_n}{a}\right) \)

Special Types:

• When \( r = -1 \): Alternating constant sequence
• When \( r = 0 \): All terms after the first are 0
• When \( 0 < r < 1 \): Terms approach 0

Geometric Series (Sum of Terms)

A geometric sequence is a number pattern in which each term is found by multiplying the previous term by a constant called the common ratio (\(r\)).

A geometric series is the sum of the terms of a geometric sequence.

General Form:

\( a,\ ar,\ ar^2,\ ar^3,\ \ldots \)

Formula for the Sum of First \(n\) Terms of a Geometric Series:

If \( r \neq 1 \), then:

\( S_n = a \frac{1 – r^n}{1 – r} \quad \text{(when } r < 1 \text{ or positive)} \) \( S_n = a \frac{r^n – 1}{r – 1} \quad \text{(when } r > 1 \text{ or negative)} \)

\( S_n \) = sum of first \(n\) terms, \( a \) = first term, \( r \) = common ratio, \( n \) = number of terms

Key Points to Remember

• Geometric sequences multiply by a fixed value each time (common ratio).
• To use the sum formula, identify \( a \), \( r \), and \( n \) clearly.
• If \( |r| < 1 \), the terms get smaller — common in fractional ratio series.
• If \( r \) is negative, terms alternate in sign (positive/negative).