Answer:
Explanation:
a) To show that a series is a geometric progression (GP), we need to check if there is a common ratio between consecutive terms. In a GP, each term is obtained by multiplying the previous term by a constant ratio.
Let's examine the given series: 8, 4, 2, ...
To find the common ratio (r), we can divide any term by its previous term:
- r = 4 / 8 = 1/2
- r = 2 / 4 = 1/2
Since the ratio between consecutive terms is the same (1/2), this series is indeed a geometric progression.
b) To find the ninth term of the series, we can use the formula for the nth term of a geometric progression:
Tn = a * r^(n-1)
Where:
- Tn is the nth term.
- a is the first term (8 in this case).
- r is the common ratio (1/2 in this case).
- n is the term number we want (9th term in this case).
Now, plug in the values:
T9 = 8 * (1/2)^(9-1)
T9 = 8 * (1/2)^8
Calculate the value:
T9 = 8 * (1/256)
T9 = 8/256
T9 = 1/32
So, the ninth term of the series is 1/32.