Question

A geometric sequence has second term 12 and fifth term 324. A) find common ratio b) calculate 10th term c) the kth term is the first term that is greater than 2000. Find the value of K

Answer 1

Answer:

a) find common ratio = 3

b) calculate 10th term = 78732

c) k = 7

Step-by-step explanation:

A geometric sequence has second term 12 and fifth term 324.

a) find common ratio

Common ratio = r

Hence

We solve for this below:

r = cube root(fifth term/second term)

r = cube root (324/12)

r = cube root (27)

r = 3

Therefore, the common ratio = 3

b) calculate 10th term

The formula for a geometric sequence is given as:

an = ar^n-1

Where

a = First term

r = Common ratio = 3

n = Nth term = 10

Step 1

We have to find the first term

Common ratio = Second term/First term

Common ratio = 3

Second term = 12

Hence:

3 = 12/x

Cross Multiply

3x = 12

x = 12/3

x = 4

Hence , First term = 4

Step 2

We find the 10th term

an = ar^n-1

a10 = 4 × 3^10 - 1

a10 = 4 × 3⁹

a10 = 4 × 19683

a10 = 78732

Therefore, the 10th term = 78732

c) the kth term is the first term that is greater than 2000. Find the value of K

For kth term,

ak = ar^k-1 >2000

a = 4, r = 3

Hence

4 × 3^k-1 > 2000

We divide both sides by 4

4 × 3^k-1/4 > 2000/4

3^k-1 > 500

We take the logarithm of both sides

log 3^k-1 > log 500

k-1 log 3 > log 500

Divide both sides by log 3

k-1 log 3/log 3 > log 500/log 3

k - 1 > 5.6567800693

k > 5.6567800693 + 1

k > 6.6567800693

k = 7

Therefore, k = 7