Answer:
[tex]k = 9[/tex]
Step-by-step explanation:
Given:
Consecutive Terms: k-5, k+7, k+55
Required:
Determine the value of k
To do this, we make use of the concept of common ratio.
The common ratio (r) of a geometric sequence is:
[tex]r = \frac{T_{n}}{T_{n-1}}[/tex]
In other words:
[tex]r = \frac{T_3}{T_2} = \frac{T_2}{T_1}[/tex]
Where 1, 2 and 3 represents the terms of the progression/sequence
So:
[tex]r = \frac{T_3}{T_2} = \frac{T_2}{T_1}[/tex] becomes
[tex]\frac{k+55}{k+7} = \frac{k+7}{k-5}[/tex]
Cross Multiply:
[tex](k+55)(k-5) = (k+7)(k+7)[/tex]
Open Brackets
[tex]k^2 + 55k - 5k - 275 = k^2 + 7k + 7k + 49[/tex]
[tex]k^2 + 50k- 275 = k^2 + 14k + 49[/tex]
Collect Like Terms
[tex]k^2 - k^2 + 50k-14k = 275 + 49[/tex]
[tex]36k = 324[/tex]
Solve for k
[tex]k = 324/36[/tex]
[tex]k = 9[/tex]
