Question

Verify the equations that appear below. Find the 150thinstance. This means find expressions for functions
a(n) and b(n) such that [a(150)]^2-[b(150)]^2= 150^3 and verify the result by showing algebraically that [a(n)]^2-[b(n)]2=n^3
for all n.

Answer 1

Doesn't look like this is the complete problem.
a(n) and b(n) are not defined.
Can you post a screenshot of the question?

Answer 2

thats whats confusing. I thought the same but that's exactly what the problem says @aum

Answer 3

sorry i did miss a part!
\[1^{2}-0^{2} = 1^3 \]
\[3^{2}-1^{2}=2^{3}\]

Answer 4

@aum

Answer 5

\[\large
~~a_n^2 - b_n^2 = n^3 \\ \large
~~1^2 - 0^2 = 1^3 \\ \large
~~3^2 - 1^2 = 2^3 \\ \large
~~6^2 - 3^2 = 3^3 \\ \large
10^2 - 6^2 = 4^3 \\
\]\[\large
n:~~ 1,~ 2,~ 3, ~~4,...\\ \large
a_n:~ 1,~ 3~, 6, 10, ... ~~~~~~~\normalsize \text {Find the nth term in this sequence.}\\ \large
b_n:~~ 0,~ 1,~ 3,~~ 6, ... ~~~~~~~\normalsize \text {Find the nth term in this sequence.} \large
\]
Find the nth term in the sequence: 1, 3, 6, 10, ....
Take the difference in terms: 2 3 4 ...
Take the difference again: 1 1
Since the second difference is a constant, this is a quadratic function of the form:
\(\large a_n = pn^2 + qn + r\) where p = 1/2 * second difference = 1/2*1=1/2.
\(\large a_n = \frac 12n^2 + qn + r\)
When \(n = 1, ~a_n = 1\):
\(\large 1 = \frac 12*1^2 + q*1 + r = \frac 12 + q + r\) --- (1)
When \(n = 2, ~a_n = 3\):
\(\large 3 = \frac 12*2^2 + q*2 + r = 2 + 2q + r\) ----- (2)
Subtract (1) from (2):
\(\large 2 = \frac 32 + q; ~~~~q = \frac 12 \)
Put q = 1/2 in (1):
\(\large 1 = \frac 12 + \frac 12 + r; ~~~~ r = 0\)

Therefore, \(\large a_n = \frac 12n^2 + \frac 12 n = \frac 12 n(n+1)\)

Similarly, \(\large b_n = \frac 12n^2 - \frac 12 n = \frac 12 n(n-1)\)

Show algebraically, \(\large a_n^2 - b_n^2 = n^3\)
\[
\large a_n^2 - b_n^2 = (a_n+b_n)(a_n-b_n) = \\ \large
\left( \frac 12n^2 + \frac 12 n + \frac 12n^2 - \frac 12 n \right ) *
\left( \frac 12n^2 + \frac 12 n - \frac 12n^2 + \frac 12 n \right ) = \\ \large
n^2 * n = n^3
\]
Find the 150th instance. That is, set n = 150 and find \(\large a_n\) and \(\large b_n\):

\(\large a_{150} = \frac 12 * 150 * 151 = 11,325\)

\(\large b_{150} = \frac 12 * 150 * 149 = 11,175\)

Make sure \(\large 11325^2 - 11175^2 = 150^3\)
\(\large 11325^2 - 11175^2 = 3375000\)
\(\large 150^3 = 3375000\)

Answer 6

wow, i had to go over this a few times for me to understand it. Thanks!