The 8th term of a go s -7/32.find it common ratio if it's first ten is 28
To find the common ratio of a geometric sequence, we need to have at least two consecutive terms. In this case, we know the first term (a) and the eighth term (t₈). However, we don't have the second consecutive term.
Let's work with the information we have:
Given:
t₈ = -7/32
a₁₀ = 28
We know that the formula for the nth term of a geometric sequence is given by:
tₙ = a * r^(n-1)
where:
tₙ is the nth term,
a is the first term,
r is the common ratio,
and n is the position of the term.
We need to find the common ratio (r) using the given information.
Step 1: Find the value of the first term (a)
To find the first term (a), we will use the given information that a₁₀ = 28. According to the formula, the 10th term is given by:
t₁₀ = a * r^(10-1)
28 = a * r^9
Step 2: Find the value of the first term (a) using the 8th term (t₈)
Substituting the given value of t₈ = -7/32 into the formula, we have:
t₈ = a * r^(8-1)
-7/32 = a * r^7
Step 3: Equate the two equations obtained in Step 1 and Step 2
Since both equations are equal to the first term (a), we can equate them:
a * r^9 = a * r^7
Step 4: Divide by 'a' on both sides
Dividing both sides by 'a', we get:
r^9 = r^7
Step 5: Divide by 'r^7' on both sides
Now, divide both sides by 'r^7' to eliminate 'r' from the exponent:
r^(9-7) = 1
r^2 = 1
Step 6: Find the value of 'r'
To find the value of 'r', we take the square root of both sides:
√(r^2) = √1
r = ±1
The common ratio can be ±1.
Therefore, the common ratio for the geometric sequence can be either 1 or -1.