The first and 5th terms of an exponential sequence are 16 and 9 respectively. Find the 7th terms of the sum of the first 7 terms

from your data:

a = 16
ar^4 = 9

16r^4 = 9
r^4 = 9/16
r^2 = ± 3/4 , but assuming r is real, r^2 = 3/4
r = ±√3 / 2

if a = 16, r = √3/2
term(7) = ar^6 = ar^2 r^4 = 16(3/4)(9/16) = 27/4

sum(7) = a(r^7 - 1)/(r-1) = a(r^6 + r^5 + r^4 + r^3 + r^2 + r + 1)

= .. you do some of this stuff

Well, it seems like you're multiplying and adding terms. Have you tried using comedy to solve math problems before? Let's give it a shot!

To find the 7th term of the sum of the first 7 terms, we'll need to do a little math dance.

The exponential sequence can be written as aₙ = a₁ * r^(n-1), where aₙ represents the nth term, a₁ is the first term, and r is the common ratio.

So, we know that the first term (a₁) is 16, and the fifth term (a₅) is 9. Let's use that information to find the common ratio (r), shall we?

a₅ = 16 * r^(5-1)
9 = 16 * r^4

Now, divide both sides by 16 to solve for r.
9/16 = r^4

Now, take the fourth root of both sides.
∛(9/16) = r

So, we now have the value of r! Let's use that information to find the seventh term (a₇), and make it laugh along the way!

a₇ = a₁ * r^(7-1)
a₇ = 16 * (∛(9/16))^(6)

And there you have it! Plug that expression into a calculator and you'll have your answer—hopefully with a chuckle or two along the way.

To find the sum of the first 7 terms of an exponential sequence, we need to know the common ratio (r). However, since the first and fifth terms are given, we can use them to find the common ratio.

The formula for the nth term of an exponential sequence is:

an = a * r^(n-1)

Where:
an = the nth term
a = the first term
r = the common ratio
n = the term number

Given that the first term (a) is 16 and the fifth term (a5) is 9, we can create two equations:

16 = a * r^(1-1) = a * r^0 = a
9 = a * r^(5-1) = a * r^4

From the first equation, we know that a = 16. We can substitute this value into the second equation:

9 = 16 * r^4

Now we can solve this equation for the common ratio (r):

r^4 = 9 / 16

Taking the fourth root of both sides:

r = (9 / 16)^(1/4)

r ≈ 0.8216

Now that we know the common ratio (r ≈ 0.8216), we can find the seventh term of the sum of the first 7 terms by substituting the values into the formula:

a7 = a * r^(7-1) = a * r^6

Substituting a = 16 and r ≈ 0.8216:

a7 = 16 * (0.8216)^6

Calculating this, we find:

a7 ≈ 6.63 (rounded to two decimal places)

Therefore, the seventh term of the sum of the first 7 terms is approximately 6.63.

To find the 7th term of the sum of the first 7 terms of an exponential sequence, we need to find the common ratio of the sequence first.

The terms of an exponential sequence can be written in the form:

a_n = a_1 * r^(n-1)

where a_n is the nth term, a_1 is the first term, r is the common ratio, and n is the position of the term in the sequence.

Given that the first term (a_1) is 16 and the 5th term is 9, we can set up two equations using the formula above:

16 = a_1 * r^(1-1) => 16 = a_1
9 = a_1 * r^(5-1) => 9 = 16 * r^4

From the first equation, we have a_1 = 16. Substituting this into the second equation:

9 = 16 * r^4

Dividing both sides by 16:

9/16 = r^4

Now we can find the value of r by taking the fourth root of both sides:

(r^4)^(1/4) = (9/16)^(1/4)

Simplifying:

r = (9/16)^(1/4)

Using a calculator, we find that r is approximately 0.842

Now we can find the 7th term of the sum of the first 7 terms. The sum of the first n terms of an exponential sequence can be found using the formula:

S_n = a_1 * (1 - r^n) / (1 - r)

where S_n is the sum of the first n terms.

For n = 7, a_1 = 16, and r = 0.842, we can substitute these values into the formula to find S_7:

S_7 = 16 * (1 - 0.842^7) / (1 - 0.842)

Calculating this expression, we find that S_7 is approximately 58.095.

Therefore, the 7th term of the sum of the first 7 terms is approximately 58.095.