91Ó°ÊÓ

For Example \(4,\) the author wanted to find a polynomial \(p\) such that $$ p(1)=1, p(2)=4, p(3)=9, p(4)=16, p(5)=31 $$ Carry out the following steps to see how that polynomial was found. (a) Note that the polynomial $$ (x-2)(x-3)(x-4)(x-5) $$ is 0 for \(x=2,3,4,5\) but is not zero for \(x=1 .\) By dividing the polynomial above by a suitable number, find a polynomial \(p_{1}\) such that \(p_{1}(1)=1\) and $$ p_{1}(2)=p_{1}(3)=p_{1}(4)=p_{1}(5)=0 $$ (b) Similarly, find a polynomial \(p_{2}\) of degree 4 such that \(p_{2}(2)=1\) and $$ p_{2}(1)=p_{2}(3)=p_{2}(4)=p_{2}(5)=0 $$ (c) Similarly, find polynomials \(p_{j},\) for \(j=3,4,5,\) such that each \(p_{j}\) satisfies \(p_{j}(j)=1\) and \(p_{j}(k)=0\) for values of \(k\) in \\{1,2,3,4,5\\} other than \(j\) (d) Explain why the polynomial \(p\) defined by $$ p=p_{1}+4 p_{2}+9 p_{3}+16 p_{4}+31 p_{5} $$ satisfies $$ p(1)=1, p(2)=4, p(3)=9, p(4)=16, p(5)=31 $$

Show that $$ \ln n<1+\frac{1}{2}+\cdots+\frac{1}{n-1} $$ for every integer \(n \geq 2\). [Hint: Draw the graph of the curve \(y=\frac{1}{x}\) in the \(x y\) -plane. Think of \(\ln n\) as the area under part of this curve. Draw appropriate rectangles above the curve.

Show that the sum of a finite arithmetic sequence is 0 if and only if the last term equals the negative of the first term.

Explain why the polynomial \(p\) defined by $$ p(x)=\frac{x^{4}-10 x^{3}+39 x^{2}-50 x+24}{4} $$ is the only polynomial of degree 4 such that \(p(1)=1\), \(p(2)=4, p(3)=9, p(4)=16,\) and \(p(5)=31\)

Show that the sum of an arithmetic sequence with \(n\) terms, first term \(b\), and difference \(d\) between consecutive terms is $$ n\left(b+\frac{(n-1) d}{2}\right) $$

Explain why an infinite sequence is sometimes defined to be a function whose domain is the set of positive integers.

Restate the symbolic version of the formula for evaluating an arithmetic series using summation notation.

Find a sequence $$ 3,-7,18,93, \ldots $$ whose \(100^{\text {th }}\) term equals \(29 .\) [Hint: A correct solution to this problem can be obtained with no calculation.]

Restate the symbolic version of the formula for evaluating a geometric series using summation notation.

Find all infinite sequences that are both arithmetic and geometric sequences.