Answer: a = 2, and r = 5
Step-by-step explanation: What we have been given here is a geometric progression. Every term in the sequence of numbers is derived by multiplying the previous term by a particular number called the common ratio, otherwise known as r. Hence if the first term is 1 for instance, the second term would be derived as 1 x r (which equals 1r), the third term would be derived as 1r x r (which equals 1r squared) and so on.
Having this in mind , we can calculate the Nth term of a geometric progression as
Nth term = a x r{to the power of n - 1}
So if we want to calculate the 4th term for instance, that would be
4th = a x r{to the power of 4 - 1} OR
4th = a x r{to the power of 3}
Similarly to calculate the 7th term would be
7th = a x r{to the power of 7 - 1}
7th = a x r{to the power of 6}
Now that we have been given the 4th (250) and 7th (31250) terms, what we now have is
a x r{to the power of 3} = 250 AND
a x r{to the power of 6} = 31250
a x r{to the power of 6}/a x r{to the power of 3} = 31250/250
After reducing both sides to their simplest form, what we now have is
r{to the power of 3} = 125
If we add the cube root sign to both sides of the equation we would have
r = 5
Having computed r as 5, we can now go back to calculate a as follows;
If a x r{to the power of 3} = 250, then
a x 125 = 250
Divide both sides of the equation by 125
a = 2
Therefore, a = 2 and r = 5
