do you see a pattern?

Question

(a +b)^4=a^4+ 4a^3 b+6a^2 + 4ab^3+b^4
(a +b)^5=a^5+5a^4 b+10a^2 b^3+5a b^4+b^4
(a +b)^6=a^6+6a^5 b+15a^4 b^2+20a^3 b^3+15a^2 b^4+6ab^5+6^6
(a +b)^7=a^7+7a^6 b+21a^4 b^2+35a^4 b^3+35a^3 b^4+21a^2 b^5+7a+b^6+b^7

Answer 1

Of course. The pattern is well known, and is manifested as the binomial distribution and Pascal's triangle.

It is well formulated mathematically. There is a formula that can tell you, for example, what the 18th term is in the expansion of (a+b)^30.

Answer 2

The coefficients can be obtained by adding coefficients from the previous expansion.

For example, the coefficients for 4 are

1 4 6 4 1

If you add these together as follows:

0+1 1+4 4+6 6+4 4+1 1+0

You get

1 5 10 10 5 1

which are the coefficients for the expansion of (a+b)^5

Also, one other thing. 4 does not divide 6 but 5 divides 10. The exponent divides all the coefficients except the initial and final ones if and only if the exponent is a prime. That is because in going from one coefficient to another, one brings in a new factor and takes out one:

1 1*4/1 = 4 4*3/2 = 6 6*2/3 = 4 4*1/4 = 1

If the exponent p is prime, there is no way a factor that divides p can take away the p that appears in the second term. So p will divide all of them. But if p is composite, such a divisor can appear; e.g., 2 dividing 4 prevents 4 from dividing the 6 above, as a 2 got divided out.

This observation was used in part by Indian mathematicians to obtain an easy method for finding if a number is prime. The idea is to use this method but to factor out the higher powers.

Answer 3

1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
just keep adding 2 terms

Answer 4

(a+b)^n in the expansion the k-th term has numerical coefficient nC(k-1). That is the pattern. If you list the terms 1 by one, you get pascals triangle.

1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1

and so on. the specialty is that when you add the k-th and k+1 th terms of a line you get the k-th term of next line.

Answer 5

(a+b)^n=a^n+nC1a^n-1b^1+nC2a^n-2b^2+nC3a^n-3b^3+...............................nCra^n-rb^r..............................+b^n
the binomial expansion