Question

the 4th and 5th terms of a GP are -13.5 and 40.5 find the first term and the sum of the first five terms​

Answer 1

First term: 0.5

Sum of the first give terms: 30.5

Explanation:

General form of a geometric sequence:

`a_n=ar^(n-1)`

where:

• 
  `a_n` is the nth term
• a is the first term
• r is the common ratio

Given terms:

• 
  `a_4=-13.5`
• 
  `a_5=40.5`

To find the common ratio r, divide consecutive terms:

`\implies r=(a_5)/(a_4)=(40.5)/(-13.5)=-3`

To find the first term, substitute the found value of r and one of the terms into the general formula:

`\begin{aligned}\implies a_5 =a(-3)^4 & = 40.5\\81a & = 40.5\\a & = (40.5)/(81)\\a & = 0.5 \end{aligned}`

Sum of the first n terms of a geometric series:

`S_n=(a(1-r^n))/(1-r)`

To find the sum of the first 5 terms of the geometric sequence, substitute n = 5 and the found values of a and r into the formula:

`\begin{aligned}\implies S_5 & =(0.5(1-(-3)^5))/(1-(-3))\\\\& =(0.5(1+243))/(1+3)\\\\& =(0.5(244))/(4)\\\\& =(122)/(4)\\\\ & = 30.5\end{aligned}`

Therefore, the sum of the first 5 terms is 30.5.