Chapter 12

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

Determine the common ratio, the fifth term, and the \(n\)th term of the geometric sequence. $$ -8,-2,-\frac{1}{2},-\frac{1}{8}, \dots $$

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

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

23-32 me the common difference, the fifth term, the \(n\)th term, and the 100th term of the arithmetic sequence. $$25,26.5,28,29.5, \dots$$

Find the \(n\)th term of a sequence whose first several terms are given. \(0,2,0,2,0,2, \ldots\)

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

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

23-32 me the common difference, the fifth term, the \(n\)th term, and the 100th term of the arithmetic sequence. $$15,12.3,9.6,6.9, \dots$$

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

Find the first six partial sums \(S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6}\) of the sequence. \(1,3,5,7, \dots\)

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

Determine the common ratio, the fifth term, and the \(n\)th term of the geometric sequence. $$ 5,5^{c+1}, 5^{2 c+1}, 5^{3 c+1}, \dots $$

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

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

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

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

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

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

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

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

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 where it fails. (a) \(p(n)=n^{2}-n+11\) is prime for all \(n\) (b) \(n^{2}>n\) for all \(n \geq 2\) (c) \(2^{2 n+1}+1\) 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 common ratio in a geometric sequence is \(\frac{2}{5},\) and the fourth term is \(\frac{5}{2} .\) Find the third term.

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

Find the first four partial sums and the \(n\)th partial sum of the sequence \(a_{n} .\) \(a_{n}=\frac{2}{3^{n}}\)

Find the term containing \(y^{3}\) in the expansion of \((\sqrt{2}+y)^{12}\)

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 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 “Midnight.” Remove some other cat (call it “Sparky”) from the group. We are left with k cats, one of whom (Midnight) is black, so by the induction hypothesis, all \(k\) of these are black. Now put Sparky back in the group and take out Midnight. 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.

The common ratio in a geometric sequence is \(\frac{3}{2},\) and the fifth term is \(1 .\) Find the first three terms.

The 20th term of an arithmetic sequence is \(101,\) and the common difference is 3 . Find a formula for the \(n\)th term.

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

Find the term containing \(b^{8}\) in the expansion of \(\left(a+b^{2}\right)^{12}\)

Which term of the geometric sequence \(2,6,18, \ldots\) is \(118,098 ?\)

Which term of the arithmetic sequence \(1,4,7, \ldots\) is 88\(?\)

Find the first four partial sums and the \(n\)th partial sum of the sequence \(a_{n} .\) \(a_{n}=\sqrt{n}-\sqrt{n+1}\)

The second and the fifth terms of a geometric sequence are 10 and \(1250,\) respectively. Is \(31,250\) a term of this sequence? If so, which term is it?

The first term of an arithmetic sequence is \(1,\) and the common difference is \(4 .\) Is \(11,937\) a term of this sequence? If so, which term is it?

Find the first four partial sums and the \(n\)th partial sum of the sequence \(a_{n} .\) \(a_{n}=\log \left(\frac{n}{n+1}\right)\) [Hint: Use a property of logarithms to write the \(n\)th term as a difference.]

Find the partial sum \(S_{n}\) of the geometric sequence that satisfies the given conditions. $$ a=5, \quad r=2, \quad n=6 $$

39-44 Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$a=1, d=2, n=10$$

Find the sum. $$\sum_{k=1}^{4} k$$

39 \(-42\) . Factor using the Binomial Theorem. $$ \begin{array}{l}{(x-1)^{5}+5(x-1)^{4}+10(x-1)^{3}+} \\\ {10(x-1)^{2}+5(x-1)+1}\end{array} $$

Find the partial sum \(S_{n}\) of the geometric sequence that satisfies the given conditions. $$ a=\frac{2}{3}, \quad r=\frac{1}{3}, \quad n=4 $$

39-44 Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$a=3, d=2, n=12$$

Find the sum. $$\sum_{k=1}^{4} k^{2}$$

39 \(-42\) . Factor using the Binomial Theorem. $$ 8 a^{3}+12 a^{2} b+6 a b^{2}+b^{3} $$

Find the partial sum \(S_{n}\) of the geometric sequence that satisfies the given conditions. $$ a_{3}=28, \quad a_{6}=224, \quad n=6 $$

39-44 Find the partial sum \(S_{n}\) of the arithmetic sequence that satisfies the given conditions. $$a=4, d=2, n=20$$