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Five Questions - Sequences, Linear equations etc.

1) Use inductive reasoning to determine the next three numbers in the pattern: 3,12,26,45.....? (I don't seem to find a pattern?)

2) Find the counterexample to show that the following statemnet is incorrect. "the sum of any odd numbers is divisible by 4. ( I tried several odd numbers and the answer is always divisible by 4)

3) Sandra drove for 234.8 miles and used 12.6 gallons of gas. estimate the number of miles Sandra's car get per gallon? ( will the answer be as simple as 234.8 / 12.6 = 18.6 miles per gallon?)

4) Sam founda number of nickles and dimes, and quarters in his room. he found thre more dimes than nickles but twice as many quartes as dimes. The total value of coins was $5.05, how many coins of each type did Sam find?

5) Use inductive reasoning find the next thre numbers in the patern: 2,9,20,35.... ( I can seem to find a pattern?)

Solution Preview

The solution file is attached.

1) using inductive reasoning to determine the next three numbers in the pattern: 3,12,26,45.....? (I don't seem to find a pattern?)

Here, we can follow the method of differences to determine the next three terms of the sequence.
First order differences:
12 - 3 = 9
26 - 12 = 14
45 - 26 = 19
Second order differences:
14 - 9 = 5
19 - 14 = 5
Since the second order differences are the same, the given sequence follows a quadratic polynomial
f(n) = an^2 + bn + c
Plug in n = 1, 2, 3 to get
f(1) = 3 = a + b + c
f(2) = 12 = 4a + 2b + c
f(3) = 26 = 9a + 3b + c
On solving this system, we get a = 5/2, b = 3/2, c = -1
 f(n) = ...

Solution Summary

Step-by-step solutions to all the five questions are provided in the solution file.

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