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
