If the 4th and 7th terms of a GP are 250 and 31250 respectively. Find the two possible values of a and r

a = 2 , r = 5

Step-by-step explanation:

The n th term of a geometric progression is

= a

where a is the first term and r the common ratio

Given the 4 th term is 250, then

ar³ = 250 → (1)

Given the 7 th term is 31250, then

a = 31250 → (2)

Dividing the 2 equations gives

= , that is

r³ = 125 ← take the cube root of both sides

r = = 5

Substitute r = 5 into (1)

a × 5³ = 250, that is

125a = 250 ( divide both sides by 125 )

a = 2

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

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