MAT 112 Quiz #10
Name: ________________________Some useful formulas.
|
A R I T H. |
The sum of a finite arithmetic sequence |
Sn = n /2 (a1 + an) |
|
The nth term of an arithmetic sequence |
an = dn + c , where c = a1 - d |
|
|
An alternative arithmetic formula |
an = a 1 + (n - 1)d |
|
|
Recursive formula, arithmetic sequence |
an+1 = an + d |
|
|
G E O M. |
The nth term of a geometric sequence |
an = a 1rn - 1 |
|
Recursive formula, geometric sequence |
an+1 = anr |
|
|
The sum of a finite geometric sequence |
Sn =a 1( (1- rn) / (1 - r) ) |
|
|
The sum of an infinite geometric sequence |
Sn =a 1 / (1 - r) |
|
|
Qd. |
The nth term of a quadratic sequence |
an = an2 + bn + c |
Pn: Sn = 5 + 7 + 9 + 11 + 13 + ... + (3 + 2n) = n(n + 4).
(7pts) Prove the proposition using the Principle of Mathematical Induction.
|
P1 is true: 3 + 2(1) = |
1(1 + 4) 5 |
|
Let Sk = 5 + 7 + 9 + ... + (3 + 2k) = |
k(k + 4), |
|
then, 5 + 7 + 9 + ... + (3 + 2k) + (3 + 2(k + 1)) = |
(k + 1)(k + 1 + 4) |
|
k(k + 4) + (3 + 2(k + 1)) = |
(k + 1)(k + 5) |
|
2 + 4k + 3 + 2k + 2 = |
k2 + 6k + 5 |
|
k2 + 6k + 5 = |
k2 + 6k + 5. |
So by the principle of mathematical induction:
Sn =5 + 7 + 9 + 11 + 13 + ... + (3 + 2n) = n(n + 4).
(3pts) Use finite differences to determine whether the sequence is arithmetic, quadratic, or neither.
|
n: |
|
1 |
|
|
|
2 |
|
|
|
3 |
|
|
|
4 |
|
|
|
5 |
|
an: |
|
5 |
|
|
|
7 |
|
|
|
9 |
|
|
|
11 |
|
|
|
13 |
|
first diff.: |
|
|
2 |
|
|
|
2 |
|
|
|
2 |
|
|
|
2 |
|
|
|
|
sec. diff.: |
|
|
|
|
0 |
|
|
|
0 |
|
|
|
0 |
|
|
|
|
|
(Bonus 2pts) Determine the sum of the first 100 terms of the sequence.
S100 = 100(100 + 4) = 10400 by our proposition.
Or using the formula for the finite sum os an arithmetic sequence:
S100 = n/2 (a1 + an) = 100/2 (5 + 3 + 2(100)) = 10400.