Video Transcript
Find the sum of the first 20 terms of a geometric sequence one, 1.07, 1.07 squared, 1.07 cubed, giving the answer to two decimal places.
The sum of the first 𝑛 terms of any geometric sequence is given by 𝑎 multiplied by one minus 𝑟 to the power of 𝑛 divided by one minus 𝑟, where 𝑎 is the first term in the sequence, 𝑟 is the common ratio, and 𝑛 is the number of terms in the sequence. In our example, the first term is one. Therefore, 𝑎 is equal to one.
The common ratio 𝑟 is equal to 1.07, as we need to multiply the first term by 1.07 to get the second term. Likewise, to get from the second to the third term, we need to multiply by 1.07 again. As we are asked to find the sum of the first 20 terms, our value for 𝑛 is equal to 20.
Substituting these three values into the formula gives us one multiplied by one minus 1.07 to the power of 20 divided by one minus 1.07. Typing the numerator into our calculator gives us negative 2.86968. The denominator one minus 1.07 is equal to negative 0.07. Dividing these two numbers gives us an answer of 40.9955. As we are asked to round our answer to two decimal places, the answer is 41 or 41.00.
The sum of the first 20 terms of the geometric sequence with 𝑎 equals one, 𝑟 equals 1.07 is 41 or 41.00.
