Understanding the G.P. Terms
In a geometric progression (G.P.), the nth term can be expressed as:
- a * r^(n-1)
where:
- a = first term
- r = common ratio
- n = term number
Given that the 4th term is x^(-k) and the 6th term is x^5, we can set up the following equations using the G.P. formula.
Setting Up the Equations
- For the 4th term:
- a * r^(4-1) = x^(-k)
- This simplifies to: a * r^3 = x^(-k)
- For the 6th term:
- a * r^(6-1) = x^5
- This simplifies to: a * r^5 = x^5
Finding the Common Ratio
Now, let's express a in terms of r:
- From the 4th term:
a = x^(-k) / r^3
- From the 6th term:
a = x^5 / r^5
Setting these equal to each other gives:
- x^(-k) / r^3 = x^5 / r^5
Cross-multiplying results in:
- x^(-k) * r^5 = x^5 * r^3
Solving for k
Dividing both sides by r^3 and x^(-k):
- r^2 = x^(5 + k)
Now consider the 10th term:
- a * r^(10-1) = a * r^9
Substituting a:
- (x^(-k) / r^3) * r^9 = x^(-k) * r^6
This means:
- x^(-k) * r^6 = x^(5 + k)
From the previous equation, we can determine:
- Thus, equating exponents allows us to find k.
Conclusion
By solving for k, we find that k = -3. This gives us the required value of k in the G.P. context.