How do you write a geometric series for which r=1/2 and n=4?

Question

Cora Alexander · Accepted Answer

#a_1, a_1/2, a_1/4, a_1/8# Where #a_1# is the first term in the series

In general the #n^(th)# term of a geometric sequence is given by: #a_n = a_"n-1"*r# Where #r# is the common ratio and #a_1# is the first term. In this example, #r=1/2# and #n=4# #a_2 = a_1* 1/2# #a_3 = a_2*1/2 = a_1*1/4# #a_4 = a_3*1/2 = a_1*1/8# In general, #a_n = a_1/2^(n-1)# Hence: the series requested is: #a_1, a_1/2, a_1/4, a_1/8# Where #a_1# is the first term in the series

Madelyn Anderson · Answer

undefined To write a geometric series where the common ratio (r) is 1/2 and the number of terms (n) is 4, you can use the formula:

$$S = a \left( \frac{1 - r^n}{1 - r} \right)$$

Where:
- $S$ represents the sum of the series,
- $a$ is the first term of the series,
- $r$ is the common ratio,
- $n$ is the number of terms.

Substitute the given values into the formula:

$$a = \text{first term}$$
$$r = \frac{1}{2}$$
$$n = 4$$

$$S = a \left( \frac{1 - \left( \frac{1}{2} \right)^4}{1 - \frac{1}{2}} \right)$$

$$S = a \left( \frac{1 - \frac{1}{16}}{1 - \frac{1}{2}} \right)$$

$$S = a \left( \frac{\frac{15}{16}}{\frac{1}{2}} \right)$$

$$S = a \left( \frac{15}{8} \right)$$

This gives you the general form of the sum of the geometric series with $r = \frac{1}{2}$ and $n = 4$, expressed in terms of the first term $a$.