Question

In a G.P the difference between the 1st and 5th term is 150, and the difference between the

2nd and the 4th terms is 48. Find the sum of the first five terms.
​

Answer 1

Either
`\displaystyle (-1522)/(√(41))` (approximately
`-238`) or
`\displaystyle (1522)/(√(41))` (approximately
`238`.)

Explanation:

Let
`a` denote the first term of this geometric series, and let
`r` denote the common ratio of this geometric series.

The first five terms of this series would be:

• 
  `a`,
• 
  `a\cdot r`,
• 
  `a \cdot r^2`,
• 
  `a \cdot r^3`,
• 
  `a \cdot r^4`.

First equation:

`a\, r^4 - a = 150`.

Second equation:

`a\, r^3 - a\, r = 48`.

Rewrite and simplify the first equation.

`\begin{aligned}& a\, r^4 - a \\ &= a\, \left(r^4 - 1\right)\\ &= a\, \left(r^2 - 1\right) \, \left(r^2 + 1\right) \end{aligned}`.

Therefore, the first equation becomes:

`a\, \left(r^2 - 1\right) \, \left(r^2 + 1\right) = 150`..

Similarly, rewrite and simplify the second equation:

`\begin{aligned}&a\, r^3 - a\, r\\ &= a\, \left( r^3 - r\right) \\ &= a\, r\, \left(r^2 - 1\right) \end{aligned}`.

Therefore, the second equation becomes:

`a\, r\, \left(r^2 - 1\right) = 48`.

Take the quotient between these two equations:

`\begin{aligned}(a\, \left(r^2 - 1\right) \, \left(r^2 + 1\right))/(a\cdot r\, \left(r^2 - 1\right)) = (150)/(48)\end{aligned}`.

Simplify and solve for
`r`:

`\displaystyle (r^2+ 1)/(r) = (25)/(8)`.

`8\, r^2 - 25\, r + 8 = 0`.

Either
`\displaystyle r = (25 - 3\, √(41))/(16)` or
`\displaystyle r = (25 + 3\, √(41))/(16)`.

Assume that
`\displaystyle r = (25 - 3\, √(41))/(16)`. Substitute back to either of the two original equations to show that
`\displaystyle a = -(497\, √(41))/(41) - 75`.

Calculate the sum of the first five terms:

`\begin{aligned} &a + a\cdot r + a\cdot r^2 + a\cdot r^3 + a \cdot r^4\\ &= -(1522√(41))/(41) \approx -238\end{aligned}`.

Similarly, assume that
`\displaystyle r = (25 + 3\, √(41))/(16)`. Substitute back to either of the two original equations to show that
`\displaystyle a = (497\, √(41))/(41) - 75`.

Calculate the sum of the first five terms:

`\begin{aligned} &a + a\cdot r + a\cdot r^2 + a\cdot r^3 + a \cdot r^4\\ &= (1522√(41))/(41) \approx 238\end{aligned}`.