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# Math Formulas: 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:

 $$a_n = a_1 \cdot q^{n-1}$$
 $$a_i = \sqrt{a_{i-1} \cdot a_{i+1}}$$
 $$S_n = \frac{a_nq - a_1}{q-1}$$
 $$S_n = \frac{a_1 \cdot \left(q^n - 1 \right)}{q-1}$$
 $$S = \frac{a_1}{1-q}, \quad (\text{for } -1 < q < 1)$$