### Video Transcript

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

Letโs start by looking at the
information that weโre given. This value of ๐ sub ๐ represents
the ๐th term of this sequence. Weโre asked to find the order of
the term whose value is 4374. That means that weโve got a
sequence, and somewhere in this sequence is this value of 4374. The order of this term means weโre
really asking, is it the second term, the 10th term, the 100th term? Thatโs what we need to find
out. We can do this by saying letโs make
the order of this term ๐, and then our ๐th 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, weโll
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, thereโs 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, weโll 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 weโre using a non-calculator
method, weโll 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.