Find the first term a and the ratio r of each geometric series.


a. ∑ k = 0 to ∞(2/3) × (1/5)ᵏ

Verified step by step guidance

1. Identify the general form of a geometric series, which is given by the sum from k = 0 to infinity of \( a \times r^k \), where \( a \) is the first term and \( r \) is the common ratio.
2. Compare the given series \( \sum_{k=0}^\infty \left( \frac{2}{3} \times \left( \frac{1}{5} \right)^k \right) \) to the general form to determine the first term \( a \) and the ratio \( r \).
3. Note that the first term \( a \) corresponds to the term when \( k = 0 \), which is \( \frac{2}{3} \times \left( \frac{1}{5} \right)^0 = \frac{2}{3} \times 1 = \frac{2}{3} \).
4. Identify the common ratio \( r \) as the factor raised to the power \( k \), which is \( \frac{1}{5} \).
5. Summarize that the first term \( a = \frac{2}{3} \) and the common ratio \( r = \frac{1}{5} \).

Key Concepts

Geometric Series Definition

A geometric series is a sum of terms where each term after the first is found by multiplying the previous term by a constant ratio r. It can be written as a + ar + ar² + ar³ + ... , where a is the first term and r is the common ratio.

Identifying the First Term (a)

The first term a of a geometric series is the initial term when the index k is zero. In the series ∑ from k=0 to ∞ of a × r^k, the first term is simply the coefficient multiplied by r raised to the zero power, which equals a.

Common Ratio (r)

The common ratio r is the factor by which each term is multiplied to get the next term. It is the base of the exponent k in the series expression a × r^k, and it determines the behavior and convergence of the series.