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Prove using Mathematical Induction
1/2 + 1/(2^2) + ... 1/(2^n) = (2^n - 1) /(2^n)
** you can use the file attached if you didnt understand how i wrote it.. its # 10
1. Determine whether the rule describes a function with the given domain and target. You must provide a specific counterexample if you determine it is not a function. (Note that the symbol 
refers to the principal (nonnegative) square root.)
a. 
R
R where 
b. ZR where 
For c, d, and e below, consider the function:
: {0,1}nZ (i.e., f maps elements from the set of all bit strings to the set of integers)
c. where 
is the position of a 1 bit
d. where 
is the position of the first 1 bit
e. where is the number of 1 bits
2. Give an example of a function from N to N that is
a. one-to-one but not onto
b. onto but not one-to-one
c. neither one-to-one nor onto
d. both onto and one-to-one, other than the identity function (this is a little tricky)
3. Suppose RR and RR where 
and 
. Find the formula for 
4. List the first six terms of the sequence defined by the recurrence relation and initial condition defined below. What do you notice about the terms of the sequence that is generated? (Extra challenges: Find a closed-form formula for f(n). Then prove, by mathematical induction, that your formula is correct.)
5. Let be defined recursively as indicated below. Find 
a. 
b. 
6. Give a recursive definition of 
, where is a non-zero real number and is a nonnegative integer.
7. Is the sequence
a solution of the recurrence relation 
if
a. 
b. 
c. 
d. 
8. Find a non-recursive (closed form) formula that generates the given sequences: 
. Indicate the domain of the formula.
a. 
b. Initial condition: 
and recurrence relation: 
9. Suppose that 
and that both and are defined for all values of . Let 
and 
. Evaluate each expression. If you do not have enough information, enter “unknown.”
a. 
b. 
c. 
d.
e. 
f.
g. 
h. 
i. 
10. Prove using Mathematical Induction.
11. Compute each of the double (nested) sums (Refer to the video lecture in Canvas.)
a. 
b. 
c. 
d. 