9. Sequences, Series, & Induction

The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a₁=4 and a_n=2a_n-1 + 3 for n≥2

In Exercises 1–12, write the first four terms of each sequence whose general term is given. a_n=(−1)ⁿ⁺¹/(2ⁿ−1)

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. 5+7+9+11+⋯+ 31

The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a₁=3 and a_n=4a_n-1 for n≥2

Write the first four terms of each sequence whose general term is given. a_n=3ⁿ

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

${\sum }_{i=4}^5(a_i/b_i{)}^3$

Write the first four terms of each sequence whose general term is given. a_n=3n+2

In Exercises 19–22, the general term of a sequence is given and involves a factorial. Write the first four terms of each sequence. a_n=2(n+1)!

The sequences in Exercises 13–18 are defined using recursion formulas. Write the first four terms of each sequence. a₁=7 and a_n=a_n-1 + 5 for n≥2

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 + 4 + 5 + ... + (n + 2) = n(n + 5)/2

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

Express each sum using summation notation. Use $1$ as the lower limit of summation and $i$ for the index of summation. $2+2^2+2^3+\cdots +2^{11}$

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

${\sum }_{i=1}^5{a}_i^2-{\sum }_{i=3}^5{b}_i^2$

Determine the first 3 terms of the sequence given by the general formula

$a_{n}=\(\frac{1}{n!+1}\)$