Pascal Triangle

We note that the coefficients (the numbers in front of each term) follow a pattern.

(a + b)^0 1
(a + b)^1 1 1
(a + b)^2 1 2 1
(a + b)^3 1 3 3 1
(a + b)^4 1 4 6 4 1
(a + b)^5 1 5 10 10 5 1
(a + b)^6 1 6 15 20 15 6 1


You can use this pattern to form the coefficients.

Notes :

- There are n + 1 terms.
- the exponent of a decrease by 1 from term to term while the exponent of b increases by 1
- __a^n +__a^(n-1)b+__a^(n-2)b^2+__a^(n-3)b^3+.............+ ___b^n


Examples :

Expand (x + 3)^4 by using the Pascal Triangle


Solution :
Step (1) : Draw a Pascal Triangle ( Refer above)

Step (2) : Create a formula of an expansion (there are n + 1 terms...so we have four terms)

(a + b)^4 = ___a^n + ___a^(n-1)b + ___a^(n-2)b^2 +___a^(n-3)b^3 + __b^4

Step (3) : Replace a = x, b = 3 and n = 4 into step 3. Also put the coefficient (refer Pascal

Triangle)on the underline in the formula

So,

(x + 3)^4 = 1x^4 + 4x^3(3) + 6x^2(3^2) + 4x(3^3) + 1(3^4)
= x^4 + 12x^3 + 54x^2 + 108x + 81