Chapter 19: Problem 4

An arithmetic sequence is given by $b, \frac{2 b}{3}, \frac{b}{3}, 0, \ldots$ (a) State the sixth term. (b) State the $k$ th term. (c) If the 20 th term has a value of 15 , find $b$.

Short Answer

Expert verified

Answer: The value of b in the given arithmetic sequence is -45/16.

Step by step solution

Calculate the common difference

In an arithmetic sequence, the common difference is constant between each consecutive pair of terms. So, we will calculate the difference between the second term and the first term, and then between the third term and the second term: Common Difference (d) = Second term - First term = $\frac{2b}{3} - b = -\frac{b}{3}$ To verify, let's calculate the difference between the third term and the second term: Difference = Third term - Second term = $\frac{b}{3}-\frac{2b}{3}=-\frac{b}{3}$ Since the common difference is the same, we can move to the next step.

Find the sixth term of the sequence

Using the arithmetic sequence formula for the nth term: $$T_n = T_1 + (n-1)d$$ Where $T_n$ is the nth term, $T_1$ is the first term, n is the term number, and d is the common difference. Since we are looking for the sixth term, $n = 6$, $T_1 = b$, and $d = -\frac{b}{3}$. $$T_6 = b + (6-1)(-\frac{b}{3}) = b - \frac{5b}{3} = -\frac{2b}{3}$$ The sixth term of the sequence is $-\frac{2b}{3}$.

Find the general formula for the kth term

Using the arithmetic sequence formula, we can write the kth term as follows: $$T_k = T_1 + (k-1)d = b+(k-1)(-\frac{b}{3})$$

Determine the value of b using the 20th term

We are given that the 20th term of the sequence is 15. We will use this information and our general formula for the kth term to determine the value of $b$. The 20th term formula is: $T_{20}=b+(20-1)(-\frac{b}{3})$ Given $T_{20}=15$, we substitute this value into the equation: $$15= b + (19)(-\frac{b}{3})$$ Now, solve for $b$: $$15 = \frac{3b}{3} - \frac{19b}{3}$$ $$\frac{45}{3} = \frac{-16b}{3}$$ Divide both sides by $\frac{-16}{3}$: $$b = -\frac{45}{16}$$ Therefore, the value of $b$ is $-\frac{45}{16}$.

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An arithmetic sequence is given by \(b, \frac{2 b}{3}, \frac{b}{3}, 0, \ldots\) (a) State the sixth term. (b) State the \(k\) th term. (c) If the 20 th term has a value of 15 , find \(b\).

Short Answer

Step by step solution

Calculate the common difference

Find the sixth term of the sequence

Find the general formula for the kth term

Determine the value of b using the 20th term

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