Explanation:
Let's assume the first term of the geometric progression is 'a', and the common ratio is 'r'.
The sum of the first and third terms can be calculated using the formula for the sum of a geometric series:
Sum = a + ar^2
According to the given information, the sum of the first and third terms is 6 1/2. So, we have the equation:
a + ar^2 = 6 1/2
Similarly, the sum of the second and fourth terms can be calculated:
Sum = ar + ar^3
According to the given information, the sum of the second and fourth terms is 9 3/4. So, we have the equation:
ar + ar^3 = 9 3/4
Now, we have a system of equations with two variables, 'a' and 'r'. We can solve these equations simultaneously to find their values.
From the first equation, we can rewrite it as:
a(1 + r^2) = 6 1/2
Similarly, from the second equation:
ar(1 + r^2) = 9 3/4
Dividing the second equation by the first equation:
(ar(1 + r^2))/(a(1 + r^2)) = (9 3/4)/(6 1/2)
Simplifying:
r = 39/26
Now, substituting this value of 'r' into the first equation:
a(1 + (39/26)^2) = 6 1/2
Simplifying further:
a(1 + (1521/676)) = 6 1/2
a(676 + 1521)/(676) = 6 1/2
a = 13/4
Therefore, the first term of the geometric progression is 13/4 or 3.25.