Geometric Sequences
Geometric Sequences - How it Works - Video
Definitions
Definitions:
Here we have the definition of a geometric sequence. The formula is ak+1= ak * r, which states that the next term is equal to the previous times a ratio.
The formula for Nth term is an= a1 * rn-1, which states the Nth term is equal to the first term times the ratio raised to power of nth term minus one.
Theorems
Example 1
Example 1:
For this example we have to use the formula, an = a1 * rn-1. The n is the number of term that we want. In this case the example wants the 1st term, the 2nd term, the 3rd term, and the 10th term.
The 1st term
=> 3 *(-1/4)1-1
=> 3 *(-1/4)0
=> 3 * 1
=> 3
The 2nd term
=> 3 *(-1/4)2-1
=> 3 *(-1/4)1
=> 3 * (-1/4)
=> (-3/4)
The 3rd term
=> 3 *(-1/4)3-1
=> 3 *(-1/4)2
=> 3 * (1/16)
=> 3/16
The 10th term
=> 3 *(-1/4)10-1
=> 3 *(-1/4)9
=> 3 * (-1/262,144)
=> -3/262,144
Example 2:
Example 2:
For this example we have to use the formula, an = a1 * rn-1. This time we have to find first term and the ratio so that we can create the formula to find any term. We have been given the 3rd term and the 6th term. Our first step is to find the ratio. Once we found the ratio, we can find our 1st term. Once we have the 1st term and the ratio we can create the formula to find any term.
Ratio
=> a6 =a1 * r5
=> 48 = (-6)/r2 * r5
=> 48 = (-6) * (r5)/(r2)
=> 48 = (-6) * (r3)
=> -8 = r3
=> r = -2
=> Write 6th term equation
=> Substitute a1 into a6 equation
=> Rearrange terms
=> Divide exponents
=> Divide by -6
=> Cube root
1st Term
=> -6 = a1 * r2
=> -6 = a1 * (-2)2
=> -6 = a1 * 4
=> -6/4 = a1
=> -3/2 = a1
=> Write a1 equation
=> Substitute -2 for the ratio
=> Square -2
=> Divide by 4
=> Simplify
Example 3:
Example 3:
We first need to find the ratio of the geometric sequence. Remember that the next term is the previous term times the ratio. So in order to find the ratio we need to divide the one term by the previous term. We should do this multiple times to see if all the ratios are the same.
Ratio
=> 0.016/0.08 = 0.2
=> 0.08/0.4 = 0.2
=> 0.4/2 = 0.2
Now that we have found the ratio, we need to use the formulas to find the sum of the first 6 terms and the sum of the entire series
Sum of First 6 terms
=> a1 * (1 - (0.2)2) / (1 - (0.2))
=> 2.49984
Sum of Series
=> 2 / (1 - (0.2))
=> 2.50