MHB How Do You Solve This Geometric Progression Problem?

[mathlearn]

If the second term is 6 and the 5th term of a geometric progression is 48.Find the first term and the common difference of it

The sum of certain number of terms of the above progression from first term is 381.Find the number of terms of it.

Any ideas on how to begin ?

 

[Greg]

"common difference" should be common ratio.

Using the given information, we may set up the equation

$$6r^3=48$$

Do you see how that equation is derived? Can you solve it for $r$?

 

[mathlearn]

"common difference" should be common ratio.

Using the given information, we may set up the equation

$$6r^3=48$$

Do you see how that equation is derived? Can you solve it for $r$?

The second term is 6 & it is given that the fifth term is 48. As geometric progression increment by the multiplication by the common ratio , the second term must be multiplied by three times the common ratio.

$$6r^3=48$$
$$r^3=8$$
$$r=2$$

Now the common ratio has been found, so now finding the first term can be eased using the formula an=arn-1

3 = a

 

[Greg]

Good work! You are correct.

$$a_n=a_{n-1}\cdot r$$

$$a_2=a_1\cdot r$$

$$a_2=2a_1$$

$$6=2a_1\implies a_1=3$$