Notation:



Number of terms in the series: n
First term: a1
Nth term: an
Sum of the first n terms: Sn
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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