Question 598796
In geometric sequences, r (known as the common ratio) is the ratio of one of the terms and the preceding term.
For the first sequence, r can be found by dividing the second term by the first term, or the third term by the second term:
a) {{{r = 12/4}}}{{{r = 3}}} or {{{r = 36/12}}}{{{r = 3}}}
The nth term is found by:
{{{a[n] = a[1]*r^(n-1)}}} where: {{{a[1]}}} is the first term, r is the common ratio, and n = the number of the term.
So for the 10th term, n = 10 and:
{{{a[10] = 4*3^(10-1)}}}
{{{a[10] = 4*3^9}}}
{{{a[10] = 4*19683}}}
{{{a[10] = 78732}}}
b) The common ratio, r, is:
{{{r = -324/972}}}
{{{r = -1/3}}} and the 10th term is:
{{{a[n] = a[1]*r^(n-1)}}} {{{a[1] = 972, {{{n = 10}}} and {{{r = -1/3}}}, so...
{{{a[10] = 972*(-1/3)^9}}}
{{{a[10] = 972*(-1/19683)}}}
{{{a[10] = -0.0493827}}} Approx.