Difference between revisions of "1976 AHSME Problems/Problem 25"
MRENTHUSIASM (talk | contribs) (Created page with "== Problem == For a sequence <math>u_1,u_2\dots</math>, define <math>\Delta^1(u_n)=u_{n+1}-u_n</math> and, for all integer <math>k>1, \Delta^k(u_n)=\Delta^1(\Delta^{k-1}(u_n...") |
MRENTHUSIASM (talk | contribs) (Finished sol.) |
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\textbf{(D) }\text{if }k=4,\text{ but not if }k=3\qquad\\ | \textbf{(D) }\text{if }k=4,\text{ but not if }k=3\qquad\\ | ||
\textbf{(E) }\text{for no value of }k</math> | \textbf{(E) }\text{for no value of }k</math> | ||
| + | |||
| + | == Solution == | ||
| + | Note that | ||
| + | <cmath>\begin{align*} | ||
| + | \Delta^1(u_n) &= \left[(n+1)^3+(n+1)\right]-\left[n^3+n\right] \\ | ||
| + | &= \left[n^3+3n^2+4n+2\right]-\left[n^3+n\right] \\ | ||
| + | &= 3n^2+3n+2, \\ | ||
| + | \Delta^2(u_n) &= \left[3(n+1)^2+3(n+1)+2\right]-\left[3n^2+3n+2\right] \\ | ||
| + | &= \left[3n^2+9n+8\right]-\left[3n^2+3n+2\right] \\ | ||
| + | &= 6n+6, \\ | ||
| + | \Delta^3(u_n) &= \left[6(n+1)+6\right]-\left[6n+6\right] \\ | ||
| + | &= \left[6n+12\right]-\left[6n+6\right] \\ | ||
| + | &= 6, \\ | ||
| + | \Delta^4(u_n) &= 6-6 \\ | ||
| + | &= 0. | ||
| + | \end{align*}</cmath> | ||
| + | Therefore, the answer is <math>\boxed{\textbf{(D)}}.</math> | ||
| + | |||
| + | More generally, we have <math>\Delta^k(u_n)=0</math> for all <math>n</math> if <math>k\geq4.</math> | ||
| + | |||
| + | ~MRENTHUSIASM | ||
| + | |||
| + | == See Also == | ||
| + | {{AHSME box|year=1976|num-b=24|num-a=26}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 11:06, 7 September 2021
Problem
For a sequence 
, define 
and, for all integer
.
If 
, then 
for all 
Solution
Note that
![\begin{align*} \Delta^1(u_n) &= \left[(n+1)^3+(n+1)\right]-\left[n^3+n\right] \\ &= \left[n^3+3n^2+4n+2\right]-\left[n^3+n\right] \\ &= 3n^2+3n+2, \\ \Delta^2(u_n) &= \left[3(n+1)^2+3(n+1)+2\right]-\left[3n^2+3n+2\right] \\ &= \left[3n^2+9n+8\right]-\left[3n^2+3n+2\right] \\ &= 6n+6, \\ \Delta^3(u_n) &= \left[6(n+1)+6\right]-\left[6n+6\right] \\ &= \left[6n+12\right]-\left[6n+6\right] \\ &= 6, \\ \Delta^4(u_n) &= 6-6 \\ &= 0. \end{align*}](http://latex.artofproblemsolving.com/8/c/5/8c57882aef7500d3c0b81018b9483bb1e4776cb0.png)
Therefore, the answer is 
More generally, we have for all if 
~MRENTHUSIASM
See Also
| 1976 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 24 |
Followed by Problem 26 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
