Question Video: Finding the Order of a Term in a Geometric Sequence given Its Value and General Term Mathematics

Find the order of the term whose value is 4,374 in the geometric sequence ๐‘Ž_๐‘› = (2/3)(3)^๐‘›.


Video Transcript

Find the order of the term whose value is 4374 in the geometric sequence ๐‘Ž sub ๐‘› equals two-thirds times three to the power of ๐‘›.

Letโ€™s start by looking at the information that weโ€™re given. This value of ๐‘Ž sub ๐‘› represents the ๐‘›th term of this sequence. Weโ€™re asked to find the order of the term whose value is 4374. That means that weโ€™ve got a sequence, and somewhere in this sequence is this value of 4374. The order of this term means weโ€™re really asking, is it the second term, the 10th term, the 100th term? Thatโ€™s what we need to find out. We can do this by saying letโ€™s make the order of this term ๐‘›, and then our ๐‘›th term will be 4374. We could then fill this into the formula and rearrange to find this value of ๐‘›, which would give us the order of this term.

We can start our rearranging by dividing both sides of this equation by two-thirds. On the left-hand side, we can recall that to divide by a fraction, we multiply by its reciprocal. And on the right-hand side, weโ€™ll be left with three to the power of ๐‘›. We can simplify the values on the left-hand side. So, we work out 2187 multiplied by three, which gives us 6561 is equal to three to the power of ๐‘›.

Now, at this stage, thereโ€™s a branch of mathematics called logarithms, which would help us solve this problem directly. But as most people learn this long after they learn about geometric sequences, weโ€™ll use a bit of trial and improvement here instead. Remember that a value like three to the power of ๐‘› equals 6561 is really equivalent to saying three to the power of what gives us this value. You might know your first powers of three off by heart, up to roughly three to the power of four equals 81. We could then continue with a few more by multiplying each of the values by three as we go up. If weโ€™re using a non-calculator method, weโ€™ll probably need to start using some pencil and paper working out. But then, we find that three to the power of eight is equal to 6561. This means that our ๐‘›-value here must be equal to eight. So, we can give our answer that the order of the term whose value is 4374 is eight, as it would be the eighth term in this sequence.

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