Question

For a given geometric sequence, the 4th term,a4 is equal to 29/625 and the 9th term is 145. Find the value of of the 13th value, a13

Answer 1

a₁₃=90,625

Explanation:

1. Find r (the common ratio of the geometric sequence):

`(29)/(625)(r)^(9-4) =145`

`r^5=((145)(625))/(29)=3125`

`r=\sqrt[5]{3125} =5`

2. Use the Recursive formula to find a₁₃:

`a_(13)=a_(12)(r)=a_(11)(r)(r)=a_(10)(r)(r)(r)=a_9(r)(r)(r)(r)`

`a_(13)=a_9(r)^4`

`a_(13)=145(5)^4=145(625)=90625`

Answer 2

`\sf a_(13) = 90625`

Explanation:

The formula for the
`\sf n`-th term (
`\sf a_n`) of a geometric sequence is given by:

`\sf a_n = a_1 \cdot r^((n-1))`

where:

-
`\sf a_n` is the
`\sf n`-th term,

-
`\sf a_1` is the first term,

-
`\sf r` is the common ratio, and

-
`\sf n` is the term number.

Given that
`\sf a_4 = (29)/(625)` and
`\sf a_9 = 145`, we can use these values to find
`\sf a_1` and
`\sf r`.

For
`\sf n = 4`, we have:

`\sf (29)/(625) = a_1 \cdot r^((4-1))`

`\sf (29)/(625) = a_1 \cdot r^3`

For
`\sf n = 9`, we have:

`\sf 145 = a_1 \cdot r^((9-1))`

`\sf 145 = a_1 \cdot r^8`

Now, we can set up a ratio between these two equations:

`\sf ((29)/(625))/(145) = (a_1 \cdot r^3)/(a_1 \cdot r^8)`

Simplify:

`\sf (29)/(625 \cdot 145) = (1)/(r^5)`

Now, solve for
`\sf r`:

`\sf r^5 = (625 \cdot 145)/(29)`

`\sf r^5 = (625 \cdot 5)/(1)`

`\sf r^5 = 3125`

`\sf r = 5`

Now that we have
`\sf r`, we can find
`\sf a_1` by substituting into one of the earlier equations. Let's use the first equation:

`\sf (29)/(625) = a_1 \cdot 5^3`

`\sf (29)/(625) = a_1 \cdot 125`

`\sf a_1 = (29)/(625 \cdot 125)`

`\sf a_1 = (29)/(78125)`

Now, we can find
`\sf a_(13)`:

`\sf a_(13) = a_1 \cdot 5^((13-1))`

`\sf a_(13) = (29)/(78125) \cdot 5^(12)`

`\sf a_(13) = (29)/(78125) \cdot 244140625`

`\sf a_(13) = 90625`

Therefore, the value of
`\sf a_(13)` is 90625.