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844 ALGEBRA. 414. In the expansion

(a+b)n = an + nan-1b + n(n-1) 1 2 an-2b2 +

we observe that in any term

(1) The exponent of b, the second term of the binomial, is one less than the number of the term from the first.

(2) The sum of the exponents is n.

(3) The last factor of the denominator of the coefficient is the same as the exponent of the second term of the binomial.

(4) The last factor of the numerator of the coefficient is the exponent of the first term of the binomial increased by 1.

Hence the (r+1)th or general term of (a+b)n is

n(n-1) (n-r+1) an-r br 1 2 3 r

Ex. Find the 6th term in the expansion of (2a+b)10. Here n = 10, and r+1=6.

We have

Note. The student should observe that the coefficient contains the same number of factors in both numerator and denominator.

EXAMPLES XXXVII. a.

Expand the following binomials ;

Write in simplest form :

10. The 4th term of (1+x)12 11. The 6th term of (2-y)8. 12. The 5th term of (a-5b)7. 13. The 15th term of (2x-1)17 14. The 7th term of 15. The 6th term of