Arithmetic and Geometric Series

Notation:

Number of terms in the series: n

First term: \(a_1\)

\(N^{th}\) term: \(a_n\)

Sum of the first n terms: \(S_n\)

Difference between successive terms: d

Common ratio: q

Sum to infinity: S

Arithmetic Series Formulas:

\( a_n = a_1 + (n-1)d \)

\( a_i = \frac{a_{i-1} + a_{i+1}}{2} \)

\( S_n = \frac{a_1 + a_n}{2} \cdot n \)

\( S_n = \frac{2 \cdot a_1 + (n-1) \cdot d}{2} \cdot n \)

Geometric Series Formulas:

\( a_n = a_1 \cdot q^{n-1} \)

\( a_i = \sqrt{a_{i-1} \cdot a_{i+1}} \)

\( S_n = \frac{a_nq – a_1}{q-1} \)

\( S_n = \frac{a_1 \cdot \left(q^n – 1 \right)}{q-1} \)

\( S = \frac{a_1}{1-q}, \quad (\text{for } -1 < q < 1) \)

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