Geometric Sequences: Explicit and Recursive Forms - College Algebra

1. A geometric sequence has the property that $a_3 = 24$ and $a_6 = -648$. If this sequence is defined recursively as $a_{n+1} = r \cdot a_n$, what is the value of the common ratio $r$?
   Answer: $-3$ In a geometric sequence, $a_6 = a_3 \cdot r^{6-3} = a_3 \cdot r^3$. Substituting the given values: $-648 = 24 \cdot r^3$, so $r^3 = \frac{-648}{24} = -27$. Taking the cube root: $r = \sqrt[3]{-27} = -3$. We can verify: if $r = -3$ and $a_3 = 24$, then $a_4 = 24(-3) = -72$, $a_5 = -72(-3) = 216$, and $a_6 = 216(-3) = -648$ ✓. Choice B gives $r^3 = 27$, which would make $a_6 = 648$ (positive). Choices C and D would give much smaller values for $a_6$.
2. The recursive formula for a geometric sequence is given by $a_1 = 5$ and $a_{n+1} = -2a_n$ for $n \geq 1$. What is the explicit formula for the $n$th term?
   Answer: $a_n = 5(-2)^{n-1}$ From the recursive formula $a_{n+1} = -2a_n$, we can identify that the common ratio is $r = -2$. The general form of a geometric sequence is $a_n = a_1 \cdot r^{n-1}$. Substituting $a_1 = 5$ and $r = -2$, we get $a_n = 5(-2)^{n-1}$. We can verify: $a_1 = 5(-2)^0 = 5$, $a_2 = 5(-2)^1 = -10$, $a_3 = 5(-2)^2 = 20$. Choice B is incorrect because it uses $(-2)^n$ instead of $(-2)^{n-1}$, giving $a_1 = 5(-2)^1 = -10$. Choice C incorrectly separates the negative sign. Choice D ignores the negative in the common ratio.
3. A geometric sequence satisfies the recursive relation $a_{n+1} = 0.8a_n$ with $a_4 = 40.96$. What is the explicit formula for this sequence?
   Answer: $a_n = 80(0.8)^{n-1}$ When you encounter a geometric sequence with a recursive relation, you need to identify the common ratio and find the first term to write the explicit formula in the standard form $a_n = a_1 \cdot r^{n-1}$. From the recursive relation $a_{n+1} = 0.8a_n$, you can see that the common ratio is $r = 0.8$. To find the first term, work backwards from the given information $a_4 = 40.96$. Since each term is obtained by multiplying the previous term by 0.8, you can find earlier terms by dividing by 0.8: $a_3 = \frac{a_4}{0.8} = \frac{40.96}{0.8} = 51.2$ $a_2 = \frac{a_3}{0.8} = \frac{51.2}{0.8} = 64$ $a_1 = \frac{a_2}{0.8} = \frac{64}{0.8} = 80$ Therefore, the explicit formula is $a_n = 80(0.8)^{n-1}$, which is choice C. Choice A uses the correct first term (80) but has the exponent as $n$ instead of $n-1$, which would make $a_1 = 80(0.8)^1 = 64$ instead of 80. Choice B incorrectly uses 64 as the coefficient, which would be $a_2$ rather than $a_1$. Choice D uses 51.2 as the coefficient, which represents $a_3$ instead of $a_1$. Remember: the standard form $a_n = a_1 \cdot r^{n-1}$ always uses the first term as the coefficient and $n-1$ as the exponent. When given information about a later term, always work backwards to find $a_1$.
4. Consider the geometric sequence defined by $a_n = 3 \cdot (\frac{1}{2})^{n-1}$. If we write this sequence recursively starting with $a_1 = k$, what is the correct recursive relationship?
   Answer: $a_1 = 3$, $a_{n+1} = \frac{1}{2}a_n$ for $n \geq 1$ When converting between explicit and recursive formulas for geometric sequences, you need to identify two key components: the first term and the common ratio that connects consecutive terms. Let's find the first term by substituting $n = 1$ into the given formula: $a_1 = 3 \cdot (\frac{1}{2})^{1-1} = 3 \cdot (\frac{1}{2})^0 = 3 \cdot 1 = 3$. So our recursive formula must start with $a_1 = 3$. For the recursive relationship, we need the common ratio. In any geometric sequence, each term equals the previous term multiplied by the common ratio. Looking at our formula $a_n = 3 \cdot (\frac{1}{2})^{n-1}$, the base $\frac{1}{2}$ is our common ratio. This means $a_{n+1} = \frac{1}{2} \cdot a_n$. Let's verify: $a_1 = 3$, $a_2 = 3 \cdot \frac{1}{2} = \frac{3}{2}$, $a_3 = \frac{3}{2} \cdot \frac{1}{2} = \frac{3}{4}$. Using our explicit formula: $a_2 = 3 \cdot (\frac{1}{2})^1 = \frac{3}{2}$ ✓. Choice A uses the wrong first term ($\frac{3}{2}$ instead of $3$) and wrong ratio ($2$ instead of $\frac{1}{2}$). Choice B has the correct ratio but wrong first term. Choice C has the correct first term but uses ratio $2$, which would make the sequence grow instead of shrink. Only choice D correctly identifies both $a_1 = 3$ and $a_{n+1} = \frac{1}{2}a_n$. Study tip: Always check your recursive formula by computing the first few terms both ways—you should get identical sequences regardless of which formula you use.
5. Consider a geometric sequence where $a_2 = 15$ and $a_5 = -405$. What is the recursive relationship for this sequence?
   Answer: $a_{n+1} = -3a_n$ When you encounter a geometric sequence problem with two given terms, your goal is to find the common ratio that connects consecutive terms through multiplication. In a geometric sequence, each term equals the previous term multiplied by the common ratio $r$. Since $a_2 = 15$ and $a_5 = -405$, you can use the relationship $a_5 = a_2 \cdot r^3$ (since you multiply by $r$ three times to go from the 2nd to 5th term). Substituting the known values: $-405 = 15 \cdot r^3$ Solving for $r$: $r^3 = \frac{-405}{15} = -27$ Therefore: $r = \sqrt[3]{-27} = -3$ The recursive relationship is $a_{n+1} = r \cdot a_n = -3a_n$, which is answer B. Let's check why the other options are incorrect: **A) $a_{n+1} = 3a_n$**: This uses $r = 3$, which would give $a_5 = 15 \cdot 3^3 = 405$, not $-405$. The sign is wrong. **C) $a_{n+1} = -\frac{1}{3}a_n$**: This uses $r = -\frac{1}{3}$, making the sequence decrease in magnitude rather than increase. **D) $a_{n+1} = \frac{1}{3}a_n$**: This uses $r = \frac{1}{3}$, which would make $a_5 = \frac{5}{9}$, far too small and positive. Study tip: Always verify your common ratio by checking that it produces the correct second given term. Calculate $a_2 \cdot r^{(n-2)}$ to find $a_n$, and pay careful attention to negative signs—they often determine the correct answer in geometric sequence problems.
6. A geometric sequence is defined recursively by $a_1 = 7$ and $a_{n+1} = ka_n$ where $k$ is a constant. If $a_5 = 112$, which explicit formula correctly represents this sequence?
   Answer: $a_n = 7 \cdot 2^{n-1}$ When you encounter a geometric sequence problem, you need to connect the recursive definition with the explicit formula. A geometric sequence has the form $a_n = a_1 \cdot r^{n-1}$ where $a_1$ is the first term and $r$ is the common ratio. First, let's find the common ratio $k$. Since $a_1 = 7$ and $a_5 = 112$, we can use the relationship $a_5 = a_1 \cdot k^4$: $112 = 7 \cdot k^4$ $k^4 = 16$ $k = 2$ Now we can write the explicit formula: $a_n = 7 \cdot 2^{n-1}$. Let's verify: $a_5 = 7 \cdot 2^{5-1} = 7 \cdot 2^4 = 7 \cdot 16 = 112$ ✓ Looking at the wrong answers: Choice A gives $a_n = 7 \cdot 2^n$, which would make $a_5 = 7 \cdot 2^5 = 224$—this formula has an extra factor of 2 because it uses $2^n$ instead of $2^{n-1}$. Choice C gives $a_n = 14 \cdot 2^{n-1}$, which would make $a_1 = 14$, contradicting the given condition that $a_1 = 7$. Choice D gives $a_n = 7 \cdot (\sqrt{2})^{n-1}$, which would make $a_5 = 7 \cdot (\sqrt{2})^4 = 7 \cdot 4 = 28$—this uses the wrong common ratio. The answer is B. Study tip: Always remember that the standard explicit formula for geometric sequences uses $r^{n-1}$, not $r^n$, because when $n=1$, you want $r^0 = 1$ to get back to $a_1$.
7. A geometric sequence has explicit formula $a_n = 48 \cdot (\frac{1}{4})^{n-1}$. Starting from this explicit form, what is the equivalent recursive definition?
   Answer: $a_1 = 48$, $a_{n+1} = \frac{1}{4}a_n$ for $n \geq 1$ When converting between explicit and recursive forms of geometric sequences, you need to identify two key components: the first term and the common ratio that connects consecutive terms. From the explicit formula $a_n = 48 \cdot (\frac{1}{4})^{n-1}$, you can find the first term by substituting $n = 1$: $a_1 = 48 \cdot (\frac{1}{4})^{1-1} = 48 \cdot (\frac{1}{4})^0 = 48 \cdot 1 = 48$. The common ratio is the base of the exponential term, which is $\frac{1}{4}$. This means each term is $\frac{1}{4}$ times the previous term, so $a_{n+1} = \frac{1}{4}a_n$. Choice C correctly identifies both components: $a_1 = 48$ and $a_{n+1} = \frac{1}{4}a_n$. Choice A has the correct first term but uses $a_{n+1} = 4a_n$, which would make the sequence increase rather than decrease. This represents the reciprocal of the actual common ratio. Choice B uses the wrong first term ($a_1 = 12$ instead of 48). This might come from incorrectly calculating $48 \cdot \frac{1}{4}$, but remember that when $n = 1$, the exponent becomes zero, not one. Choice D has the correct recursive relationship but starts with $a_1 = 192$. This appears to come from multiplying 48 by 4, perhaps confusing the direction of the ratio. Study tip: Always verify your recursive formula by computing the first few terms both ways. The explicit formula $a_n = a_1 \cdot r^{n-1}$ directly shows you $a_1$ and $r$, which become the building blocks of your recursive definition.

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