9. Sequences, Series, & Induction

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

In Exercises 10–11, express each sum using summation notation. Use i for the index of summation. 1/3 + 2/4 + 3/5 + ... + 15/17

In Exercises 8–9, find each indicated sum. This is a summation, refer to the textbook.

Evaluate $\frac{40!}{4!38!}$

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: 3 is a factor of n³ - n.

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

$\frac{200!}{198!}$

Evaluate each factorial expression. (n+2)!/n!

Use mathematical induction to prove that each statement is true for every positive integer n. ${\sum }_{i=1}^{n}5\cdot 6^i=6(6^n-1)$

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. $1^2+2^2+3^2+\cdots +{15}^2$

Use mathematical induction to prove that each statement is true for every positive integer n. 2 + 4 + 8 + ... + 2ⁿ = 2ⁿ⁺¹ - 2

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

$\frac{20!}{300}$

Find each indicated sum. ${\sum }_{k=1}^{5}k(k+4)$

Express each sum using summation notation. Use a lower limit of summation of your choice and k for the index of summation. a+ar+ar²+⋯+ ar¹²

Find each indicated sum. ${\sum }_{i=1}^5\frac{i!}{(i-1)!}$