91Ó°ÊÓ

Use the Principle of Mathematical Induction to show that the given statement is true for all natural numbers \(n\). $$ n(n+1)(n+2) \text { is divisible by } 6 $$

Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=0 ; \quad r=\frac{1}{7} $$

Expand each expression using the Binomial Theorem. $$ \left(x^{2}-y^{2}\right)^{6} $$

The given pattern continues. Write down the nth term of a sequence \(\left\\{a_{n}\right\\}\) suggested by the pattern. \(\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \ldots\)

Extended Principle of Mathematical Induction The Extended Principle of Mathematical Induction states that if Conditions I and II hold, that is, (I) A statement is true for a natural number \(j\). (II) If the statement is true for some natural number \(k \geq j\), then it is also true for the next natural number \(k+1\). then the statement is true for all natural numbers \(\geq j\). Use the Extended Principle of Mathematical Induction to show that the number of diagonals in a convex polygon of \(n\) sides is \(\frac{1}{2} n(n-3)\) [Hint: Begin by showing that the result is true when \(n=4\) (Condition I).]

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 8th term is 4; 18th term is - 96

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 15th term is 0 ; 40th term is -50

Find the first term and the common difference of the arithmetic sequence described. Find a recursive formula for the sequence. Find a formula for the nth term. 14th term is \(-1 ; 18\) th term is -9

Expand each sum. \(\sum_{k=1}^{n}(k+1)^{2}\)

Find each sum. The sum of the first 120 terms of the sequence $$ 14,16,18,20, \ldots $$