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### Mathematical Investigation

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# Mathematical Investigation

By | August 2013
Page 1 of 3
QUEZON HILL NATIONAL HIGH SCHOOL

A Mathematical Investigation

by Gey-steve F. Ana-ao

July 28, 2013

STAGE 1. PRELIMINARY SKIRMISHING
Construct a 1 unit square. Investigate

Formulated Questions:
2. If adjacent unit squares are added, How many unit squares are there all in all?

STAGE 2. GESTATING
Focus of Investigation: The number of 1 unit squares formed when adjacent squares are drawn on all of its sides.

STAGE 3. EXPLORING SYSTEMATICLLY
“n” times the adjacent squares are added to the 1 unit square| Illustration| Number of 1 unit squares(Sn)| 1|
| 5|
2|

| 13|
3|

| 25|
4|

| 41|
“n” times the adjacent squares are added to the 1 unit square| Total number of 1 unit squaresSn| Pattern| 1| 5| 1+4(1)|
2| 13| 1+4(1)+4(2)|
3| 25| 1+4(1)+4(2)+4(3)|
4| 41| 1+4(1)+4(2)+4(3)+4(4)|
5| 61| 1+4(1)+4(2)+4(3)+4(4)+4(5)|
n| Sn| 1+4(1)+4(2)…+4(n)|

STAGE 4: MAKING CONJECTURE
N| 1| 2| 3| 4| 5|
sn| 5| 13| 25| 42| 61|

First Difference
8 12 16 20
Second Difference
4 4 4

Since the second difference are equal, then Sn is Quadratic in nature.
This can be solved using the finite difference.
1. )2a = 42.) 3a+b=83.) a + b + c = 5
a = 2...

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