Fiona gives birth to triplets. The triplets grow up to each have their own set of triplets. These sets of triplets go on to have their ?

Question

own triplets as well. If this pattern continues throughout many generations, how many generations have there been to generate a total of 9840 descendants of Fiona?

Don't count Fiona as your first term.

How can I do this using a geometric sequence?

Answer 1

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The number of kids in each generation is the geometric sequence starting with 3 and each future term is multiplied by 3.

She has 3 kids.  Each of the 3 had 3 kids so there are 9, etc.

Using this equation for the generic form of the n'th term of a geometric sequence:

a(n) = ar^(n - 1)

Where a is the first term (3) and r is the common ratio (3), we get:

a(n) = 3 * 3^(n - 1)

We can simplify this since we have the product of two numbers of the same base which is the same as the sum of the exponents:

a(n) = 3^(1 + n - 1)

a(n) = 3n

The equation for the sum of the first n terms of a geometric sequence is:

S(n) = a(1 - r^n) / (1 - r)

Again, a = 3 and r = 3, so:

S(n) = 3(1 - 3^n) / (1 - 3)

S(n) = 3(1 - 3^n) / (-2)

Distribute the 3 and simplify:

S(n) = (3 - 3 * 3^n) / (-2)

S(n) = [3 - 3^(n + 1)] / (-2)

S(n) = -[3 - 3^(n + 1)] / 2

S(n) = [3^(n + 1) - 3] / 2

How many generations are needed to have a total of 9840 descendents?  Set S(n) to 9840 and solve for n:

9840 = [3^(n + 1) - 3] / 2

19680 = 3^(n + 1) - 3

19683 = 3^(n + 1)

19683 is a power of 3, so:

3⁹ = 3^(n + 1)

Two values that are equal with the same bases so the exponents must also be equal:

9 = n + 1

8 = n

It will take 8 generations to have that many descendents.

Source(s): https://www.varsitytutors.com/hotmath/hotmath_help...

Answer 2

After 1 generation you have 3 descendants.

After 2 generations you have 9 additional descendants

After 3 generations you have 27 additional descendants.

3, 9, 27, 81, ...

This is a geometric sequence with a first term of 3 and a common ratio of 3.

a = 3

r = 3

The formula for the sum of the first n terms of a geometric sequence is:

S[n] = a(1 - r^n) / (1 - r)

Plug in your values of a and r:

S[n] = 3(1 - 3^n) / (1 - 3)

S[n] = 3(1 - 3^n) / -2

You want that to equal 9840 descendants:

3(1 - 3^n) / -2 = 9840

3(1 - 3^n) = -19680

1 - 3^n = -19680/3

1 - 3^n = -6560

-3^n = -6560 - 1

-3^n = -6561

3^n = 6561

You could just try some integer values (guess and check) to figure out n. Or you could use logs. Take the log of both sides:

log(3^n) = log(6561)

Use this rule of logs --> log(a^b) = b log(a)

n log(3) = log(6561)

n = log(6561) / log(3)

n = 8

Double-check:

3^8 = 6561

Also:

3 + 9 + 27 + 81 + 243 + 729 + 2187 + 6561 = 9840

Answer:

8 generations

Source(s): https://www.mathsisfun.com/algebra/sequences-sums-...