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Pascal's triangle

Pascal's triangle shows the coefficients of the binomial theorem with negative exponents

as geometric progressions.

 

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Explanation

We're going to describe Pascal's triangle for the development of binomial coefficients with negative exponents. A number is the sum of the number to the left and right there just above.

−8																	1		−8		36		−120		330		···
−7																1		−7		28		−84		210		−462
−6															1		−6		21		−56		126		−252		···
−5														1		−5		15		−35		70		−126		210
−4													1		−4		10		−20		35		−56		84		···
−3												1		−3		6		−10		15		−21		28		−36
−2											1		−2		3		−4		5		−6		7		−8		···
−1										1		−1		1		−1		1		−1		1		−1		1
0									1
1								1		1
2							1		2		1
3						1		3		3		1
4					1		4		6		4		1
5				1		5		10		10		5		1
6			1		6		15		20		15		6		1
7		1		7		21		35		35		21		7		1
8	1		8		28		56		70		56		28		8		1

You will notice, that for negative exponents each row has infinitely many terms. These have alternately positive and negative coefficients that get bigger and bigger. They form the coefficients of geometric sequences

(1 + x)⁻¹ = 1 − x + x² − x³ + x⁴ − x⁵ + x⁶ − x⁷ + ···

(1 + x)⁻² = 1 − 2x + 3x² − 4x³ + 5x⁴ − 6x⁵ + 7x⁶ − 8x⁷ + ···

 

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