9. Sequences, Series, & Induction

Write the first four terms of each sequence whose general term is given. a_n=2n/(n+4)

Find each indicated sum. ${\sum }_{i=0}^4\frac{(-1{)}^i}{i!}$

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

Use mathematical induction to prove that each statement is true for every positive integer n. 3 + 7 + 11 + ... + (4n - 1) = n(2n + 1)

Use a calculator's factorial key to evaluate each expression. (300/20)!

Write the first 6 terms of the sequence given by the recursive formula $a_{n}=a_{n-2}+a_{n-1}$ ; $a_1=1$ ; $a_2=1$.

Use mathematical induction to prove that each statement is true for every positive integer n. n + 2 > n

Use mathematical induction to prove that each statement is true for every positive integer n. 6 is a factor of n(n + 1)(n + 2).

A statement Sn about the positive integers is given. Write statements S_k and S_k+1 simplifying statement S_k+1 completely. Sn: 3 + 7 + 11 + ... + (4n - 1) = n(2n + 1)

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1/2+2/3+3/4+⋯+ 14/(14+1)

Use mathematical induction to prove that the statement is true for every positive integer n. 5 + 10 + 15 + ... + 5n = (5n(n+1))/2

A statement S_n about the positive integers is given. Write statements S_k and S_k+1 simplifying statement S_k+1 completely. Sn: 4 + 8 + 12 + ... + 4n = 2n(n + 1)

Evaluate each factorial expression. 17!/15!