Geometric sequences and series
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This document discusses geometric sequences and series. It defines a geometric sequence as a sequence where each term after the first is found by multiplying the preceding term by a common ratio. It provides formulas for finding the nth term of a geometric sequence and the nth partial sum, where the nth term is the first term multiplied by the common ratio to the power of n-1, and the nth partial sum is the first term multiplied by 1 minus the common ratio to the nth power, divided by 1 minus the common ratio.
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Geometric sequences and series
- 2. Common Ratio A geometric sequence is a sequence in which each term after the fisrt term is a common multiple of the preceding term. The common multiple is called the common ratio. The common ratio, r, in any geometric sequence can be found by dividing by term, except the first, by the preceding term.
- 3. Find the nth term of a Geometric Sequence In general, a geomtric sequence with the first term, a1, and a common ratio, r, has the following terms: a1, a1r, a1r2, a1r3, a1r4,…, a1rn-1, Thus, we can see that the nth ter of a geomtric sequence is given by the following formula: 𝑎 𝑛 = 𝑎1 𝑟 𝑛−1
- 4. Find the nth Partial Sum of a Geometric Sequence The following formula is used to get the nth partial sum of a geometric sequence: 𝑠 𝑛 = 𝑎1(1−𝑟 𝑛) 1−𝑟 , r≠1