Question Video: Finding the Value of a Certain Term of a Geometric Sequence given Its General Term Mathematics

Video Transcript

Find the value of the second term of the geometric sequence 𝑎 sub 𝑛 equals one-sixth times two to the power of 𝑛 plus three, where 𝑛 is greater than or equal to one.

Let’s remember that a geometric sequence is a sequence where the ratio between consecutive terms is constant. Each of the terms in the sequence is referenced by an index 𝑛. We’re told here that 𝑛 is greater than or equal to one. Here, the first term of this sequence will have an index of one, and we can say that this is the term 𝑎 sub one. The second term is 𝑎 sub two, the third term is 𝑎 sub three, and so on.

Given that we have an 𝑛th term formula here, we can find any term value in the sequence by filling in that value of 𝑛. For example, if we wanted to find the 100th term, we would substitute 𝑛 equals 100 into the 𝑛th term formula.

In this question, however, we’re looking for the second term. That’s the term with index two. And so we substitute 𝑛 equals two into the 𝑛th term. This gives us 𝑎 sub two equals one-sixth times two to the power of two plus three. Two to the power of two plus three can be simplified further as two to the power of five. Two to the power of five can be evaluated as 32. And multiplying that by one-sixth, we have 32 over six. And of course, we can simplify this fraction by taking out a common factor of two from the numerator and the denominator.

And so the second term of this sequence is 16 over three.