...those terms in the expansion of (1 +ж)", in which the index of ж is a multiple of 3, is 799. Find the sum of the squares of the coefficients in the expansion of (1 +ж)", when n is a positive integer. 800. Find the sum of the products of every two consecutive coefficients...

...(11), prove that 1.а„ + o1om+1 + a2am+i + ..... + amaim 2s"». (im- T) (4m- 3) . . . . 3.1 14. Find the sum of the squares of the coefficients in the expansion of (1 + a:)*-. 15. The third, fourth, and fifth terms of the expanded binomial (x 4- y)", are а, Ъ, с,...

...be the coefficient of the (r+i)th term of (i— x)~m, show that wv-f (»i+ i)r_,= (m+ i),. 41. Find the sum of the squares of the coefficients in the expansion of (i +#)", when n is a positive integer. NB Equate the coeflfts. of J?B in (i + *)n.(^+ i)" and in (l...

...prove that the coefficients of the second term, and of the last but one, are each я. 6. Show that the sum of the squares of the coefficients in the expansion of (a+b)a by the binomial theorem = 1.2.3 (2я) (1.2.3... я)2 ' 7. A person who receives an annuity of...

...expansion of (1 + xfr, then <„ + <, + i, + = (1 - 4ж)Ч 40. Write down the sum of 1 1.3 1.3.5 41. Find the sum of the squares of the coefficients in the expansion of (1 + x)", where n is a positive integer. ,, т„ 1.3.5......(2r-l) a. 4. 6 that (1-х)' 43. Prove that...

...and multiply the two series together ; then the term which does not contain x will be This shews that the sum of the squares of the coefficients in the expansion of (1 + x)n is equal to the coefficient of the term which does not contain x in / l\n (l + :e)8n the expansion...

...Shew that if tr denote the middle term in the expansion 41. Write down the sum of 1 1.3 1.3.5 42. Find the sum of the squares of the coefficients in the expansion of (1 + x)", where и is a positive integer. 43. If pr = """ — jr — - , prove that 44. Shew that if m...

...6 + 62-g5+68-6ir+etc= 6 + -0833333--0011574 + -0000267--0000007 + etc. = 6-0822019 + etc. (5.) Find the sum of the squares of the coefficients in the expansion of (1 +ж)", n being a positive integer. Put also (1 + ж)" = Ax" + Bж"-i + Cж'-2 + ____ + Cж2 + Bгc...

...multiply the two series together ; then the term which does not contain ж will be This shews that the sum of the squares of the coefficients in the expansion of (l+ж)" is equal to the coefficient of the term which does not contain x in (l\n (l + ж)2ft 1 + -...

...prove that the index, when rational, must be of the form q* — 2, where q is an integer. 54. Shew that the sum of the squares of the coefficients in the expansion of (1 + x + of)", where n is a. positive integer, is 0 \r \r\2ñ^¿r' 55. Shew that, if n is any positive...