if term 6=-243 and term 3=72 of a geometric sequence determine the constant ratio
term6 = ar^5 = -243
term3 = ar^2 = 72
divide:
r^3 = -243/72 = -27/8
take cube root
r = -3/2
perfect
To determine the constant ratio of a geometric sequence, we can use the formula:
\[\text{{term}}_n = \text{{term}}_1 \times \text{{ratio}}^{(n-1)}\]
Given that term 6 is -243 and term 3 is 72, we can set up two equations:
For term 6:
\[-243 = \text{{term}}_1 \times \text{{ratio}}^{(6-1)}\]
For term 3:
\[72 = \text{{term}}_1 \times \text{{ratio}}^{(3-1)}\]
We can solve these equations simultaneously to find the constant ratio. Let's solve them step by step:
Step 1: Divide the two equations to eliminate the term 1:
\[\frac{-243}{72} = \frac{\text{{term}}_1 \times \text{{ratio}}^{(6-1)}}{\text{{term}}_1 \times \text{{ratio}}^{(3-1)}}\]
Simplifying, we get:
\[-3.375 = \text{{ratio}}^{(6-1)-(3-1)}\]
Step 2: Simplify the exponents:
\[-3.375 = \text{{ratio}}^{5-2}\]
Step 3: Simplify the exponent:
\[-3.375 = \text{{ratio}}^3\]
Step 4: Find the cube root of both sides:
\[\sqrt[3]{-3.375} = \sqrt[3]{\text{{ratio}}^3}\]
Step 5: Solve for the ratio:
\[\text{{ratio}} = \sqrt[3]{-3.375}\]
Thus, the constant ratio of the geometric sequence is approximately -1.5.
To determine the constant ratio of a geometric sequence, we need to use the formula:
Term(n) = Term(1) * (Ratio)^(n-1)
Given that Term 6 equals -243 and Term 3 equals 72, we can set up two equations:
-243 = Term(1) * (Ratio)^(6-1)
72 = Term(1) * (Ratio)^(3-1)
Let's solve these equations step by step:
Equation 1: -243 = Term(1) * (Ratio)^5
Equation 2: 72 = Term(1) * (Ratio)^2
To eliminate the Term(1) variable, we can divide Equation 1 by Equation 2:
(-243/72) = [(Term(1) * (Ratio)^5)] / [(Term(1) * (Ratio)^2)]
Simplifying further, we get:
(-243/72) = (Ratio)^3
Now, we can calculate the value of the constant ratio (Ratio) by taking the cube root of (-243/72):
Ratio = (cube root of (-243/72))
Calculating the value results in:
Ratio ≈ -3
Therefore, the constant ratio of the geometric sequence is -3.