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Chapter 2: Problem 37

Which of the following statements is NOT true about the arithmetic sequence \(17,12,7,2, \cdots ?\) A. The fifth term is \(-3\) . B. The sum of the first 5 terms is 35 . C. The eighth term is \(-18\) . D. The common difference of consecutive terms is \(-5\) . E. The common ratio of consecutive terms is \(-5\) .

Short Answer

Expert verified
A. The 5th term is -3. B. The sum of the first five terms is 35. C. The 8th term is -18. D. The common difference of consecutive terms is -5. E. The common ratio of consecutive terms is -5. Answer: E. The common ratio of consecutive terms is -5 (NOT true).

Step by step solution

01

Determine the Common Difference

To find the common difference, subtract the second term from the first term, or the third term from the second term, and so on. In this case, the common difference is \(12 - 17 = -5\) or \(7 - 12 = -5\).
02

Determine the nth term formula

To find any term in an arithmetic sequence, we use the formula: \(a_n = a_1 + (n-1)d\), where \(a_n\) is the nth term, \(a_1\) is the first term, n is the term number, and d is the common difference. In this case, \(a_1 = 17\) and \(d = -5\), so the formula becomes: \(a_n = 17 + (n-1)(-5)\).
03

Verify Statement A

Statement A says the fifth term is \(-3\). To verify this, we can substitute n=5 into the nth term formula we derived in step 2: \(a_n = 17 + (5-1)(-5) = 17 - 20 = -3\). Thus, statement A is true.
04

Verify Statement B

Statement B says the sum of the first five terms is \(35\). To find the sum of the first n terms in an arithmetic sequence, we use the formula: \(S_n = \frac{n}{2}(a_1 + a_n)\), where \(S_n\) is the sum of the first n terms, \(a_1\) is the first term, and \(a_n\) is the nth term. First, we need to find the fifth term, so substituting \(n=5\) into the nth term formula: \(a_5 = 17 + (5-1)(-5) = -3\), which we found in step 3. Now, we can substitute the values into the sum formula: \(S_5 = \frac{5}{2}(17 + (-3)) = \frac{5}{2}(14) = 35\). Thus, statement B is true.
05

Verify Statement C

Statement C says the eighth term is \(-18\). To verify this, substitute n=8 into the nth term formula from step 2: \(a_n = 17 + (8-1)(-5) = 17 - 35 = -18\). Thus, statement C is true.
06

Verify Statement D

Statement D says the common difference of consecutive terms is \(-5\). We already determined the common difference in step 1, which is indeed \(-5\). Thus, statement D is true.
07

Verify Statement E

Statement E says the common ratio of consecutive terms is \(-5\). However, the arithmetic sequence does not have a common ratio, as it is not a geometric sequence. A geometric sequence has a constant ratio between consecutive terms, while an arithmetic sequence has a constant difference. Thus, statement E is NOT true. So the correct answer is: E. The common ratio of consecutive terms is \(-5\) (NOT true).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Common Difference
The concept of 'common difference' in an arithmetic sequence is fundamental and quite simple to understand. Put plainly, it is the consistent interval or gap between any two consecutive terms in the sequence. For students grappling with sequences, identifying the common difference is a crucial step because it characterizes the entire sequence.

In our example with the series beginning at 17 and descending, the common difference is found by subtracting any term from the preceding term, like so: 12 - 17 which equals -5. This negative common difference tells us that our sequence is decreasing rather than increasing.
Nth Term Formula
Understanding the 'nth term formula' is akin to having a secret key that unlocks any door in a treasure-filled castle. In arithmetic sequences, this miraculous key is the formula to find any term in the sequence: \(a_n = a_1 + (n-1)d\). Here, \(a_n\) represents the term you are seeking, \(a_1\) is your starting block, the very first term, \(n\) counts the footsteps or term number, and \(d\) continues to be your faithful companion, the common difference.

Let's consider our example: the formula \(a_n = 17 + (n-1)(-5)\) can magically provide the value for any term you wish to find in the given sequence. By simply substituting the term number for \(n\), the nth term comes to light with ease.
Sequence Sum Formula
Lastly, let's discuss the 'sequence sum formula'. Just as a baker uses a recipe to sum up the ingredients for a cake, this formula sums up the terms of an arithmetic sequence. It is particularly useful when the sequence is longer and manual addition becomes cumbersome. Nevertheless, the formula is \(S_n = \frac{n}{2}(a_1 + a_n)\), where \(S_n\) represents the sum of the first \(n\) terms, \(a_1\) remains the first term, and \(a_n\) is the nth term.

For the example provided, to calculate the sum of the first 5 terms, we found the 5th term to be \(a_5 = -3\) and plugged the values into our formula: \(S_5 = \frac{5}{2}(17 + (-3)) = 35\). This straightforward approach yields the sum swiftly and accurately without the need for adding each term individually.

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The braking distance, \(y\) feet, for Damon's car to come to a complete stop is modeled by \(y=\frac{3\left(x^{2}+10 x\right)}{40},\) where \(x\) is the speed of the car in miles per hour. According to this model, which of the following is the maximum speed, in miles per hour, Damon can be driving so that the braking distance is less than or equal to 150 feet? \(\mathbf{F} .10\) \(\mathbf{G} .30\) \(\mathbf{H} .40\) \(\mathbf{J} .50\) \(\mathbf{K} .60\)

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