given the geometric sequences:9 90/17, 900/289
find an explicit formula for an, where the first term is a1=9
an=
a10=
to find r, (90/17) / 9 = 10/17
and (900/289) / (90/17) = 10/17
a = 9
r = 10/17
term(n) = 9(10/17)^(n-1)
term(10) = 9(10/17)^9 = appr .07589
To find the explicit formula for an in a geometric sequence, we can use the formula:
an = a1 * r^(n-1)
Where:
- a1 is the first term of the sequence,
- r is the common ratio, and
- n is the position of the term (n = 1, 2, 3, ...).
For the given geometric sequence with a1 = 9, let's find the common ratio (r) by dividing each term by its previous term:
r1 = (90/17) / 9 = 10/17
r2 = (900/289) / (90/17) = 10/17
Since r1 = r2, we can conclude that the common ratio (r) is 10/17.
Now, we can find the explicit formula for the sequence:
an = a1 * r^(n-1)
an = 9 * (10/17)^(n-1)
So, the explicit formula for an is:
an = 9 * (10/17)^(n-1)
To find the value of a10, substitute n = 10 into the explicit formula:
a10 = 9 * (10/17)^(10-1)
a10 = 9 * (10/17)^9
a10 ≈ 4.596
Therefore, the explicit formula for an is an = 9 * (10/17)^(n-1), and the value of a10 is approximately 4.596.