Arithmetic Sequences: Explicit and Recursive Forms - College Algebra

An arithmetic sequence starts with $a_1 = 15$ and has a common difference of $d = -3$ . For which value of $n$ does the explicit formula $a_n = 15 - 3(n-1)$ first give a negative result?
Answer: $n = 7$ Set $a_n < 0$ : $15 - 3(n-1) < 0$ . This gives $15 < 3(n-1)$ , so $5 < n-1$ , which means $n > 6$ . The first integer value where $n > 6$ is $n = 7$ . Checking: $a_6 = 15 - 3(5) = 0$ and $a_7 = 15 - 3(6) = -3$ . Choice A gives $a_5 = 3 > 0$ . Choice B gives $a_6 = 0$ . Choice D is too late since $a_7$ is already negative.
A student writes two forms for the same arithmetic sequence: Explicit form: $a_n = 8 + 3(n-1)$ and Recursive form: $a_1 = k$ , $a_n = a_{n-1} + d$ for $n \geq 2$ . If both forms represent the same sequence, what are the values of $k$ and $d$ ?
Answer: $k = 8, d = 3$ When you encounter questions comparing explicit and recursive forms of arithmetic sequences, you need to identify the first term and common difference from each form, then match them up. The explicit form $a_n = 8 + 3(n-1)$ follows the standard pattern $a_n = a_1 + d(n-1)$ , where $a_1$ is the first term and $d$ is the common difference. Comparing these, you can see that $a_1 = 8$ and $d = 3$ . You can verify this by finding $a_1 = 8 + 3(1-1) = 8 + 0 = 8$ . The recursive form states $a_1 = k$ and $a_n = a_{n-1} + d$ . Since both forms represent the same sequence, the first term $k$ must equal 8, and the common difference $d$ must equal 3. Looking at the wrong answers: Choice A switches the values, giving $k = 3$ and $d = 8$ . This likely comes from misreading the explicit form's structure. Choice B gives $k = 5$ , which might result from incorrectly calculating $a_1$ as $8 + 3 - 1 = 10$ or some other computational error, but keeps the correct common difference. Choice C correctly identifies $k = 8$ but incorrectly states $d = 4$ , perhaps by misreading the coefficient in the explicit form. Therefore, $k = 8$ and $d = 3$ , making D correct. Study tip: Always extract the first term and common difference systematically from explicit forms using the pattern $a_n = a_1 + d(n-1)$ , and double-check by computing $a_1$ directly when $n = 1$ .
An arithmetic sequence has $a_1 = 7$ and $a_{15} = 91$ . What is the recursive formula for this sequence?
Answer: $a_1 = 7$ ; $a_n = a_{n-1} + 6$ for $n \geq 2$ To find the common difference, use $a_n = a_1 + (n-1)d$ . With $a_{15} = 91$ and $a_1 = 7$ : $91 = 7 + (15-1)d$ , so $84 = 14d$ , giving $d = 6$ . The recursive formula is $a_1 = 7$ and $a_n = a_{n-1} + 6$ for $n \geq 2$ . Choice B uses the first term as the common difference. Choice C uses the number of steps between terms. Choice D uses the total difference between the given terms.
Consider the arithmetic sequence defined by $a_n = 5n - 2$ . Which of the following correctly expresses this sequence in recursive form?
Answer: $a_1 = 3$ ; $a_n = a_{n-1} + 5$ for $n \geq 2$ When converting between explicit and recursive forms of arithmetic sequences, you need to identify two key components: the first term and the common difference (the amount added between consecutive terms). Let's work with the given explicit formula $a_n = 5n - 2$ . First, find the first term by substituting $n = 1$ : $a_1 = 5(1) - 2 = 3$ . Next, determine the common difference by calculating a few terms. We have $a_1 = 3$ , $a_2 = 5(2) - 2 = 8$ , and $a_3 = 5(3) - 2 = 13$ . The difference between consecutive terms is $8 - 3 = 5$ and $13 - 8 = 5$ , so the common difference is 5. The recursive form states that each term equals the previous term plus the common difference: $a_1 = 3$ and $a_n = a_{n-1} + 5$ for $n \geq 2$ . This matches option D. Option A has the correct first term but uses a common difference of 2 instead of 5. Option B incorrectly calculates $a_1 = 5$ (this would be $a_2$ ) and subtracts 2, which would create a decreasing sequence. Option C uses $a_1 = -2$ , which would be $a_0$ if we extended the sequence backward, and while it has the correct common difference of 5, the wrong starting point makes the entire recursive definition incorrect. Study tip: Always verify your recursive form by computing the first few terms and checking they match the explicit formula. The coefficient of $n$ in an arithmetic sequence's explicit form is always the common difference.
An arithmetic sequence has the property that $a_5 = 17$ and $a_9 = 29$ . If this sequence is defined recursively starting with $a_1$ , what is the value of $a_1$ ?
Answer: $a_1 = 5$ First find the common difference: $d = \frac{a_9 - a_5}{9 - 5} = \frac{29 - 17}{4} = 3$ . Using $a_5 = a_1 + 4d$ : $17 = a_1 + 4(3) = a_1 + 12$ , so $a_1 = 5$ . Choice A results from using the wrong relationship between terms. Choice C comes from confusing the position number with the term value. Choice D results from incorrectly calculating the common difference.
A sequence is defined recursively as $a_1 = -3$ and $a_n = a_{n-1} + 4$ for $n \geq 2$ . Which explicit formula correctly represents the 12th term of this sequence?
Answer: $a_{12} = -3 + 11 \cdot 4 = 41$ For an arithmetic sequence with $a_1 = -3$ and common difference $d = 4$ , the explicit formula is $a_n = a_1 + (n-1)d$ . For the 12th term: $a_{12} = -3 + (12-1) \cdot 4 = -3 + 11 \cdot 4 = 41$ . Choice A incorrectly uses $n$ instead of $(n-1)$ . Choice C treats it as exponential instead of arithmetic. Choice D incorrectly applies the formula structure.
An arithmetic sequence has its 4th term equal to 22 and its 10th term equal to 46. What is the explicit formula for the $n$ th term of this sequence?
Answer: $a_n = 10 + 4(n-1)$ When you encounter an arithmetic sequence problem with two given terms, you need to find the common difference first, then use it to determine the explicit formula. In an arithmetic sequence, the difference between consecutive terms is constant. Given that $a_4 = 22$ and $a_{10} = 46$ , you can find the common difference by recognizing that from the 4th term to the 10th term, there are 6 steps (since 10 - 4 = 6). The total change is 46 - 22 = 24, so the common difference is $d = \frac{24}{6} = 4$ . Now you can work backwards to find the first term. Since $a_4 = 22$ and each term decreases by 4 as you go backwards: $a_3 = 18$ , $a_2 = 14$ , and $a_1 = 10$ . The explicit formula is $a_n = 10 + 4(n-1)$ , which is choice C. Let's check why the other options fail. Choice A gives $a_1 = 4$ , which would make $a_4 = 4 + 4(3) = 16$ , not 22. Choice B gives $a_1 = 6$ , making $a_4 = 6 + 4(3) = 18$ , still incorrect. Choice D uses a different reference point but gives $a_1 = 22 + 4(-3) = 10$ , which seems right, but when you check $a_{10}$ , you get $22 + 4(6) = 46$ —this works too! However, D isn't in standard form. Study tip: Always verify your formula by plugging in the given term positions. The standard explicit formula form $a_n = a_1 + d(n-1)$ is usually expected, so convert any alternative forms to this structure.
The explicit formula for an arithmetic sequence is $a_n = 23 - 5(n-1)$ . What are the first three terms of this sequence, and what is the recursive definition?
Answer: First three terms: 23, 18, 13; Recursive: $a_1 = 23$ , $a_n = a_{n-1} - 5$ From $a_n = 23 - 5(n-1)$ : $a_1 = 23 - 5(0) = 23$ , $a_2 = 23 - 5(1) = 18$ , $a_3 = 23 - 5(2) = 13$ . The common difference is $d = -5$ , so the recursive formula is $a_1 = 23$ and $a_n = a_{n-1} - 5$ . Choice B incorrectly starts with $a_2$ . Choice C and D incorrectly use positive 5 as the common difference.

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