Chapter 13

\(25-32\) . Find the \(n\) th term of a sequence whose first several terms are given. $$ 1, \frac{3}{4}, \frac{5}{9}, \frac{7}{16}, \frac{9}{25}, \dots $$

Find the first three terms in the expansion of \((x+2 y)^{20}\)

\(F_{n}\) denotes \(n t h\) term of the Fibonacci sequence discussed in Section \(13.1 .\) Use mathematical induction to prove the statement. \(F_{3 n}\) is even for all natural numbers \(n\)

\(27-36\) . Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence. $$ 11,8,5,2, \dots $$

Determine the common ratio, the fifth term, and the nth term of the geometric sequence. $$ 1, \sqrt{2}, 2,2 \sqrt{2}, \ldots $$

\(25-32\) . Find the \(n\) th term of a sequence whose first several terms are given. $$ \frac{3}{4}, \frac{4}{5}, \frac{5}{6}, \frac{6}{7}, \dots $$

Find the first four terms in the expansion of \(\left(x^{1 / 2}+1\right)^{30}\)

An Annuity That Lasts Forever An annuity in perpetuity is one that continues forever. Such annuities are useful in setting up scholarship funds to ensure that the award continues. (a) Draw a time line (as in Example 1\()\) to show that to set up an annuity in perpetuity of amount \(R\) per time period, the amount that must be invested now is \(A_{p}=\frac{R}{1+i}+\frac{R}{(1+i)^{2}}+\frac{R}{(1+i)^{3}}+\cdots+\frac{R}{(1+i)^{n}}+\cdots\) where \(j\) is the interest rate per time period. (b) Find the sum of the infinite series in part (a) to show that $$ A_{p}=\frac{R}{i} $$ (c) How much money must be invested now at 10\(\%\) per year, compounded annually, to provide an annuity in perpetuity of \(\$ 5000\) per year? The first payment is due in one year. (d) How much money must be invested now at 8\(\%\) per year, compounded quarterly, to provide an annuity in perpetuity of \(\$ 3000\) per year? The first payment is due in one year.

\(27-36\) . Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence. $$ -12,-8,-4,0, \dots $$

Determine the common ratio, the fifth term, and the nth term of the geometric sequence. $$ 144,-12,1,-\frac{1}{12}, \dots $$

\(25-32\) . Find the \(n\) th term of a sequence whose first several terms are given. $$ 0,2,0,2,0,2, \dots $$

Find the last two terms in the expansion of \(\left(a^{2 / 3}+a^{1 / 3}\right)^{25}\)

\(F_{n}\) denotes \(n t h\) term of the Fibonacci sequence discussed in Section \(13.1 .\) Use mathematical induction to prove the statement. $$ F_{1}^{2}+F_{2}^{2}+F_{3}^{2}+\cdots+F_{n}^{2}=F_{n} F_{n+1} $$

Amortizing a Mortgage When they bought their house, John and Mary took out a \(\$ 90,000\) mortgage at 9\(\%\) interest, repayable monthly over 30 years. Their payment is \(\$ 724.17\) per month (check this, using the formula in the text). The bank gave them an amortization schedule, which is a table showing how much of each payment is interest, how much goes toward the principal, and the remaining principal after each payment. The table below shows the first few entries in the amortization schedule. After 10 years they have made 120 payments and are wondering how much they still owe, but they have lost the amortization schedule. (a) How much do John and Mary still owe on their mortgage? [Hint: The remaining balance is the present value of the 240 remaining payments. (b) How much of their next payment is interest, and how much goes toward the principal? IHint: Since \(9 \% \div 12=0.75 \%\) , they must pay 0.75\(\%\) of the remaining principal in interest each month.

\(27-36\) . Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence. $$ \frac{7}{6}, \frac{5}{3}, \frac{13}{6}, \frac{8}{3}, \dots $$

Determine the common ratio, the fifth term, and the nth term of the geometric sequence. $$ -8,-2,-\frac{1}{2},-\frac{1}{8}, \ldots $$

\(25-32\) . Find the \(n\) th term of a sequence whose first several terms are given. $$ 1, \frac{1}{2}, 3, \frac{1}{4}, 5, \frac{1}{6}, \dots $$

Find the first three terms in the expansion of $$ \left(x+\frac{1}{x}\right)^{40} $$

\(F_{n}\) denotes \(n t h\) term of the Fibonacci sequence discussed in Section \(13.1 .\) Use mathematical induction to prove the statement. $$ F_{1}+F_{3}+\cdots+F_{2 n-1}=F_{2 n} $$

\(27-36\) . Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence. $$ 25,26.5,28,29.5, \dots $$

Determine the common ratio, the fifth term, and the nth term of the geometric sequence. $$ 3,3^{5 / 3}, 3^{7 / 3}, 27, \dots $$

\(33-36\) . Find the first six partial sums \(S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\) of the sequence. $$ 1,3,5,7, \ldots $$

Find the middle term in the expansion of \(\left(x^{2}+1\right)^{18}\)

\(27-36\) . Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence. $$ 15,12.3,9.6,6.9, \ldots $$

Determine the common ratio, the fifth term, and the nth term of the geometric sequence. $$ t, \frac{t^{2}}{2}, \frac{t^{3}}{4}, \frac{t^{4}}{8}, \dots $$

\(33-36\) . Find the first six partial sums \(S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\) of the sequence. $$ 1^{2}, 2^{2}, 3^{2}, 4^{2}, \ldots $$

Find the fifth term in the expansion of \((a b-1)^{20}\)

\(27-36\) . Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence. $$ 2,2+s, 2+2 s, 2+3 s, \dots $$

Determine the common ratio, the fifth term, and the nth term of the geometric sequence. $$ 1, s^{2 / 7}, s^{4 / 7}, s^{6 / 7}, \dots $$

\(33-36\) . Find the first six partial sums \(S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\) of the sequence. $$ \frac{1}{3}, \frac{1}{3^{2}}, \frac{1}{3^{3}}, \frac{1}{3^{4}}, \dots $$

Find the 24 th term in the expansion of \((a+b)^{25}\)

\(27-36\) . Determine the common difference, the fifth term, the nth term, and the 100 th term of the arithmetic sequence. $$ -t,-t+3,-t+6,-t+9, \ldots $$

Determine the common ratio, the fifth term, and the nth term of the geometric sequence. $$ 5,5^{c+1}, 5^{2 c+1}, 5^{3 c+1}, \ldots $$

\(33-36\) . Find the first six partial sums \(S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\) of the sequence. $$ -1,1,-1,1, \dots $$

Find the 28 th term in the expansion of \((A-B)^{30}\)

Find and prove an inequality relating 100\(n\) and \(n^{3}\) .

The tenth term of an arithmetic sequence is \(\frac{55}{2},\) and the second term is \(\frac{7}{2}\). Find the first term.

The first term of a geometric sequence is \(8,\) and the second term is 4. Find the fifth term.

\(37-40\) . Find the first four partial sums and the nth partial sum of the sequence \(a_{m}\) $$ a_{n}=\frac{2}{3^{n}} $$

Find the 100 th term in the expansion of \((1+y)^{100}\)

True or False? Determine whether each statement is true or false. If you think the statement is true, prove it. If you think it is false, give an example in which it fails. (a) \(p(n)=n^{2}-n+11\) is prime for all \(n .\) (b) \(n^{2}>n\) for all \(n \geq 2\) . (c) \(n^{2 n+1}+1\) is divisible by 3 for all \(n \geq 1\) (d) \(n^{3} \geq(n+1)^{2}\) for all \(n \geq 2\) . (e) \(n^{3}-n\) is divisible by 3 for all \(n \geq 2\) (f) \(n^{3}-6 n^{2}+11 n\) is divisible by 6 for all \(n \geq 1\)

The 12 th term of an arithmetic sequence is \(32,\) and the fifh term is 18 . Find the 20 \(\mathrm{th}\) term.

The first term of a geometric sequence is \(3,\) and the third term is \(\frac{4}{3} .\) Find the fifth term.

\(37-40\) . Find the first four partial sums and the nth partial sum of the sequence \(a_{m}\) $$ a_{n}=\frac{1}{n+1}-\frac{1}{n+2} $$

All Cats Are Black? What is wrong with the following "proof" by mathematical induction that all cats are black? Let \(P(n)\) denote the statement "In any group of \(n\) cats, if one cat is black, then they are all black." Step 1 The statement is clearly true for \(n=1\) Step 2 Suppose that \(P(k)\) is true. We show that \(P(k+1)\) is true. Suppose we have a group of \(k+1\) cats, one of whom is black; call this cat "Tadpole." Remove some other cat (call it "Sparky") from the group. We are left with \(k\) call it Sparky" (Tadpole) is black, so by the induction hypothesis, all \(k\) of these are black. Now put Sparky back in the group and take out Tadpole. We again have a group of \(k\) cats, all of whom-except possibly Sparky - are black. Then by the induction hypothesis, Sparky must be black too. So all \(k+1\) cats in the original group are black. Thus by induction \(P(n)\) is true for all \(n .\) since everyone has seen at least one black cat, it follows that all cats are black.

Find the second term in the expansion of $$ \left(x^{2}-\frac{1}{x}\right)^{25} $$

The 100 th term of an arithmetic sequence is \(98,\) and the comon difference is \(2 .\) Find the first three terms.

The common ratio in a geometric sequence is \(\frac{2}{5},\) and the fourth term is \(\frac{5}{2} .\) Find the third term.

\(37-40\) . Find the first four partial sums and the nth partial sum of the sequence \(a_{m}\) $$ a_{n}=\sqrt{n}-\sqrt{n+1} $$

Find the term containing \(x^{4}\) in the expansion of \((x+2 y)^{10}\)