9. Sequences, Series, & Induction

Write the first four terms of each sequence whose general term is given. a_n=(−1)ⁿ(n+3)

Find each indicated sum. ${\sum }_{i=5}^{9}11$

Use mathematical induction to prove that each statement is true for every positive integer n. 1 + 3 + 5 + ... + (2n - 1) = n²

In Exercises 81–85, use a calculator's factorial key to evaluate each expression.

$\frac{54!}{(54-3)!3!}$

In Exercises 55–60, express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. $a+(a+d)+(a+2d)+\cdots +(a+nd)$

Use mathematical induction to prove that each statement is true for every positive integer n. 2 is a factor of n² - n.

In Exercises 61–68, use the graphs of and to find each indicated sum.

${\sum }_{i=1}^5(a_i+3b_i)$

In Exercises 5–10, a statement Sn about the positive integers is given. Write statements S_k and S_k+1 simplifying statement S_k+1 completely. Sn: 2 is a factor of n² - n + 2.

The first 4 terms of a sequence are $\(\left\[\lbrace\]\sqrt\)3,2\(\sqrt\)3,3\(\sqrt\)3,4\(\sqrt\)3,\(\ldots\[\right\]\rbrace\)$. Continuing this pattern, find the $7^{\(\th\)}$ term.

Find each indicated sum. ${\sum }_{i=1}^{6}5i$

Write the first four terms of each sequence whose general term is given. $a_n=1/(n-1)!$

Use mathematical induction to prove that the statement is true for every positive integer n. 1 + 4 + 4^2 + ... + 4^(n-1) = ((4^n)-1)/3

A statement S_n about the positive integers is given. Write statements S₁, S₂ and S₃ and show that each of these statements is true. S_n: 1 + 3 + 5 + ... + (2n - 1) = n²