9. Sequences, Series, & Induction

Express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1+2+3+⋯+ 30

Find each indicated sum. ${\sum }_{i=1}^4{(-\frac{1}{2})}^i$

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

${\sum }_{i=1}^5(a_i^2+1)$

In Exercises 43–54, express each sum using summation notation. Use 1 as the lower limit of summation and i for the index of summation. 1+3+5+⋯+ (2n−1)

Find each indicated sum. ${\sum }_{i=1}^{4}2i^2$

Use mathematical induction to prove that each statement is true for every positive integer n. 1 + 2 + 2² + ... + 2^n-1 = 2^{n - 1}

Write the first four terms of each sequence whose general term is given. a_n = 7n - 4

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

${\sum }_{i=4}^5(a_i/b_i{)}^2$

Use mathematical induction to prove that each statement is true for every positive integer n. (ab)ⁿ = aⁿ bⁿ

Use mathematical induction to prove that each statement is true for every positive integer n. 4 + 8 + 12 + ... + 4n = 2n(n + 1)

Write the first five terms of the sequence whose first term is 9 and whose general term is

for n≥2.

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

${\sum }_{i=1}^5(2a_i+b_i)$

Express each sum using summation notation. Use $1$ as the lower limit of summation and $i$ for the index of summation. $4+4^2/2+4^3/3+\cdots +4^n/n$

Use a calculator's factorial key to evaluate each expression. 20!/(20−3)!