Question

Find an explicit rule for the nth term of a geometric sequence where the second and fifth terms are -36 and 2304, respectively.

Answer 1

@jdoe0001

Answer 2

so to get from the 2nd term to the 5th term
we have to multiply by the same number "r", 3 times
once to get the 3rd term
again to get the 4th term
and lastly to get the 5th term

so \(\begin{array}{llll}
term&value
\\\hline\\
a_2&36\\
a_3&36\cdot r\\
a_4&(36\cdot r)\cdot r\\
a_5&(36\cdot r\cdot r)\cdot r\to 2,304
\end{array}\)

so we can say that \(\bf 36r\cdot r\cdot r=2,304\implies 36r^3=2,304\implies r=\cfrac{2,304}{36}\)

Answer 3

hmm actually I missed the 3 =) heheh lemme fix that

Answer 4

\(\bf 36r\cdot r\cdot r=2,304\implies 36r^3=2,304\implies r^3=\cfrac{2,304}{36}
\\ \quad \\
\implies\large r=\sqrt[3]{\cfrac{2,304}{36}}\)

Answer 5

is that complete?

Answer 6

well... no... .but what would that give us for "r"?

Answer 7

oh, let me solve.

Answer 8

r=4

Answer 9

so my answer for this problem would be
an = 9 • 4^(n - 1) ?

Answer 10

yeap so if r = 4 and the 2nd term is 36
that means that 4 * "something" is 36
and that "something" is our first term

so our first term will be \(\bf "something"\cdot 4=36\implies "something"=\cfrac{\cancel{ 36 }}{\cancel{ 4 }}\to 9\leftarrow a_1\)

so now we know our \(\bf a_1\ and \ r\) so just plug them in :) \(\Large a_{\color{brown}{ n}}=a_1\cdot r^{{\color{brown}{ n}}-1}\)