Final answer:
To find the common ratio and fifth term of a geometric progression, set up equations using the given information, solve them simultaneously, and substitute values to find the common ratio and fifth term. The common ratio is found to be 2, and the fifth term is 128.
Step-by-step explanation:
To find the common ratio and fifth term of a geometric progression, we can use the given information and the properties of a geometric progression. Let's start by setting up two equations using the given information:
First, we know that the sum of the first and third terms is 40. Let the first term be 'a' and the common ratio be 'r'. So, the sum of the first and third term can be written as:
a + ar^2 = 40
Second, we know that the fourth and sixth terms are in the ratio 1:4. This means that:
ar^3 / ar^5 = 1/4
Now, we can solve these equations simultaneously to find the values of 'r' and 'a'. Using the first equation:
a + ar^2 = 40
Factoring out 'a' gives:
a(1 + r^2) = 40
Dividing both sides by '1 + r^2' gives:
a = 40 / (1 + r^2)
Now, substituting 'a' in the second equation:
(40 / (1 + r^2))(r^3) / (40 / (1 + r^2))(r^5) = 1/4
Simplifying this equation gives:
r^3 / r^5 = 1/4
Combining the powers of 'r' gives:
r^-2 = 1/4
Taking the reciprocal of both sides gives:
r^2 = 4
Taking the square root of both sides gives:
r = ±2
Since a geometric progression cannot have a negative common ratio, we discard the negative value and conclude that the common ratio is 2.
Finally, to find the fifth term, we can substitute 'r' into the first equation and solve for 'a'.
Using 'r = 2', the first equation becomes:
a + 4a = 40
Combining like terms gives:
5a = 40
Dividing both sides by 5 gives:
a = 8
Now, we can find the fifth term by using the formula for the nth term of a geometric progression:
T_5 = a * r^(5-1) = 8 * 2^4 = 8 * 16 = 128
Therefore, the common ratio is 2 and the fifth term is 128. So, the correct answer is option: d. Common ratio: 2, Fifth term: 128