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Writing the nth Term of a Recursive Sequence In Exercises \(53-56,\) write the first five terms of the sequence defined recursively. Use the pattern to write the nth term of the sequence as a function of \(n\). $$a_{1}=6, \quad a_{k+1}=a_{k}+2$$

Use a graphing utility to graph the first 10 terms of the sequence. $$a_{n}=20(-1.25)^{n-1}$$

Find the sum using the formulas for the sums of powers of integers. $$\sum_{j=1}^{10}\left(3-\frac{1}{2} j+\frac{1}{2} j^{2}\right)$$

Finding the Probability of a Complement You are given the probability that an event will happen. Find the probability that the event will not happen. $$P(E)=\frac{2}{3}$$

Writing the nth Term of a Recursive Sequence In Exercises \(53-56,\) write the first five terms of the sequence defined recursively. Use the pattern to write the nth term of the sequence as a function of \(n\). $$a_{1}=25, \quad a_{k+1}=a_{k}-5$$

Find the indicated \(n\)th partial sum of the arithmetic sequence. $$-6,-2,2,6, \ldots, \quad n=50$$

Evaluate \(_{n} C_{r}\) using a graphing utility. \(_{10} C_{7}\)

Use a graphing utility to graph the first 10 terms of the sequence. $$a_{n}=2(1.3)^{n-1}$$

Writing the nth Term of a Recursive Sequence In Exercises \(53-56,\) write the first five terms of the sequence defined recursively. Use the pattern to write the nth term of the sequence as a function of \(n\). $$a_{1}=81, \quad a_{k+1}=\frac{1}{3} a_{k}$$

Find the indicated \(n\)th partial sum of the arithmetic sequence. $$4.2,3.7,3.2,2.7, . . ., \quad n=12$$