evng .my teacher gave me this as a homework but i dnt undertand.
1.In a GP the 3rd term is equal to four times the first term and the sixth term is equal to three times the fourth term plus 32. Determine the sequence.
Good evening! I can help you understand how to solve this homework problem.
Let's break down the information given step by step:
1. "In a GP" - This means that the given sequence is a Geometric Progression (GP), where each term is found by multiplying the previous term by a constant number called the common ratio (r).
2. "The 3rd term is equal to four times the first term" - Let's assume that the first term of the GP is 'a' and the common ratio is 'r'. This tells us that the third term, which is at position n = 3, is equal to 4 times the first term (n = 1). Mathematically, this can be written as: a * r^2 = 4 * a.
3. "The sixth term is equal to three times the fourth term plus 32" - Similarly, let's use the same assumption for the first term and common ratio. This tells us that the sixth term (n = 6) is equal to three times the fourth term (n = 4) plus 32. Mathematically, this can be written as: a * r^5 = 3 * (a * r^3) + 32.
Now, we have two equations:
Equation 1: a * r^2 = 4 * a
Equation 2: a * r^5 = 3 * (a * r^3) + 32
To solve these equations, we can use substitution or elimination method.
Substituting the value from Equation 1 into Equation 2:
(4 * a) * r^5 = 3 * ((4 * a) * r^3) + 32
4 * r^5 = 3 * 4 * r^3 + 32
4 * r^5 = 12 * r^3 + 32
Now we have an equation with only one variable, r. Solving this equation will give us the value of the common ratio.
Once we have the common ratio, we can substitute it back into Equation 1 to find the value of the first term, 'a'.
To determine the sequence, we can use the value of 'a' and 'r' to find the subsequent terms of the GP by multiplying the common ratio to each term.
I hope this explanation helps you understand how to approach the problem. Let me know if you have any further questions!