Question

The first term of a geometric sequence is 6 and the fourth term is 384. What is the sum of the first 10 terms of the corresponding series?​

Answer 1

2097150

Explanation:

GIVEN :-

• First term of G.P. = 6
• Forth term of G.P. = 384

TO FIND :-

• Sum of first 10 terms of the G.P.

CONCEPT TO BE USED IN THIS QUESTION :-

Geometric Progression :-

• It's a sequence in which the successive terms have same ratio.

• General form of a G.P. ⇒ a , ar , ar² , ar³ , ....... [where a = first term ; r = common ratio between successive terms]

• Sum of 'n' terms of a G.P. ⇒
  `(a(r^n - 1))/(r-1)`.

[NOTE :-
`(a(1-r^n))/(1-r)` can also be the formula for "Sum of n terms of G.P." because if you put 'r' there (assuming r > 0) you'll get negative value in both the numerator & denominator from which the negative sign will get cancelled from the numerator & denominator. YOU'LL BE GETTING THE SAME VALUE FROM BOTH THE FORMULAES.]

SOLUTION :-

Let the first term of the G.P. given in the question be 'a' and the common ratio between successive terms be 'r'.

⇒ a = 6

It's given that forth term is 384. So from "General form of G.P." , it can be stated that :-

`=> ar^3 = 384`

`=> 6r^3 = 384`

Divide both the sides by 6.

`=> (6r^3)/(6) = (384)/(6)`

`=> r^3 = 64`

`=> r = \sqrt[3]{64} = 4`

Sum of first 10 terms
`= (6(4^(10)-1))/(4 - 1)`

`= (6(1048576 - 1))/(3)`

`= 2 * 1048575`

`= 2097150`