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777. Required the coefficient of x5 in the expansion of (a—x)9. 778. Required the coefficient of x in the expansion of (5a3 —4x3)7. 779. Required the coefficient of x12 in the expansion of (a3 —b3x2) 1⁄2 780. Required the 6th term of the expansion of √4a2cx- 3c3y3. 781. Required the 5th term of the expansion of (ax—by) ̄ 782. Required the 6th term of the expansion of (ax-bx2 √−1)3• 783. Required the 8th term of the expansion of (−s+t√ — 1)−4. 784. Required the greatest term in the expansions of

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785. Required the middle terms of (a”—x”)1o and of (1+x)2n, which latter may be made to assume the form

1.3.5...(2n-1)

1.2.3...n

(2x)n.

786. Write down the (r+1)th terms of (3a+2x)- and of

(xy-3yz).

2n

787. Expand (2+1) and (x-1)*, bracketing in pairs

the terms equidistant from the beginning and end of each

sion.

expan

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793. a−(a+b)n+(a+2b)n.” — 1 — (a+3b)n.

2

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n-In-2

795. If the coefficients of the expansion of (a-x)m be multiplied by 1", 2", 3′′, &c., in order, the result will =0 when m is greater than n.

796. If the coefficients of (a-x)" be multiplied by 1.2, 2.3, 3.4, &c., in order, the result will=o.

797. If a be greater than a, show that the sum of all the terms of the expansion of (x+a)", after the first two, is less than (2n-1).ax—1.

798. If z2+z+1 =0, show that the sum of those terms in the expansion of (1+x)", in which the index of a is a multiple of 3, is

= { (1 + 2) " + (1 + x)" + (1 + x2)"}.

799. Find the sum of the squares of the coefficients in the expansion of (I+)", when n is a positive integer.

800. Find the sum of the products of every two consecutive coefficients in the expansion of (1+x)", n being a positive integer. 801. Prove that 4 times the product of the sums of the odd and even terms of the expansion of (a+x)" is = (a+x) 2n — (a — x)2n. 802. Find the coefficient of a" in the expansion of

(1+x+2x2+3×3 + . . .)2.

803. If a, b, c, d be any consecutive coefficients of an expanded binomial, then will (bc+ad) (b−c)=2(ac2 — db2).

804. If the (r+ 1)th term of the expansion of (1+x)" when n is a positive integer be equal to the (r+ 3)th term, it is required to find r. 805. If s=sum of two quantities, p= their product, and q=quotient, prove that p2= 4.5.6

p2=s^(q2—
q2-493 +4.5.q+. 595 +

1.2 1.2.3

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and the sum of the nth powers of the two quantities s"-np.s"-2

+

n(n−3).p2.sn

1.2

806. Given P and Q, the coefficients of the pth and qth terms of the expansion of (a+x)"; find n.

807. If n, be the coefficient of the (r+1)th term of (1+x)" expanded, show that (n+p),=n,+n-1 P1 + N r - 2 P2 + ... + n1 Pr-x + Pr. 808. If m, be the coefficient of the (r+ 1)th term of the expansion of (1-x), show that m,+ (m + 1),, = (m + 1),.

-m

809. If x=a(1+h), where h is a very small fraction; find the

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811. Find the value of {a— ~a”—x"}÷x" when x=0.

812. Find the value of {x2— a2)2 + (x− a) } ÷ { (1+x− a)3 — 1 }

when x=a.

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814. If I be a very small quantity, show that

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815. Required the coefficient of a+ in the expansion of (1+x+x2). 816. Required the term involving a2bc3 in the expansion of (a+b+c)7.

817. Find the coefficient of x5 in the expansion of

(a + bx + cx2 + dx3)6.

818. In the expansion of (2+4x-3x2+x3)7, find the coefficient of x8.

m

819. In the expansion of (a-bx-ca2)", find the coefficient of x5. 820. Required the coefficient of an-5.b.c3 in the expansion of (a+b+c)".

821. Find the term involving b2.c3.ef in the expansion of (a+2b+3c+4d+5e+6f)1o.

822. Find the coefficient of x5 in (1+2x+3x2 +4x3 +. . .)6 expanded.

823. Find the coefficient of x in (a+bx+cx2+...)” expanded. 824. Expand (1 + x + x2 + x3 + ...) 1⁄2.

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828. Required the coefficient of a" in (1+2x+3x2+...)2 expanded. 829. What is the number of terms in the expansion of (a+b+c)"? 830. What is the number of terms in the expansion of (a + bx + cx2+dx3) 4?

831. Find the middle term in the expansion of (1+x-x2)n. 832. What is the coefficient of the middle term in the expansion of (2-5x-7x2 + x3 +3x+)5 ?

INDETERMINATE COEFFICIENTS.

833. Resolve

6x2-4x-6

(x — 1) (x − 2) (x − 3)

into partial fractions.

834. Find the fractions which, when united by addition, shall

give

x2
(x2 — 1)(x-2)

835. Find the fractions whose sum is

1+5x+3x2

(1 + x)2 (1 + 2x)2

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843. If y=ax+bx2 + cx3 + dx++ ..., express x in a series of ascending powers of y.

844. If ny3—xy=n, expand y in a series of ascending powers of x. 845. If y3+x=3y, expand y in a series of ascending powers of x.

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+... to infinity.

I

I

I

+

+

...

+ to n terms, and

1.2.3 2.3.4 3.4.5

851. Find the sum of 12+62+152+202+152+62+12. 852. Sum the series 12+11+552+1652+3302+4622+4622 +3302+1652+552 + 112 + 12.

853. Sum the series 1.9 +9.36.+36.84+84.126+1262+126.84 +84.36+36.9+9.1.

SCALES OF NOTATION.

854. Express the denary number 790158 in the septenary scale. 855. Express the quinary number 34402 in the quaternary scale. 856. Transform 6587 and 3907 from the common scale to the duodenary, and find the product of the results.

857. Divide 14332216 by 6541 in the septenary scale.

858. Divide 95088918 by tt 4 in the duodenary scale.

859. Find the radix of that scale in which 40501 is equivalent to the denary number 5365.

860. The number 124 in the denary scale is expressed by 147 in another scale; required the radix of the latter.

861. Any number is divisible by 8, if the number consisting of the last 3 digits in order be divisible by 8.

862. Any number consisting of an even number of digits, in a system whose radix is r, is divisible by r+1, if the digits equidistant from each end are the same.

863. If N, N' be any two numbers in the denary scale, composed of the same digits differently arranged, prove that Ñ-N' is divisible by 9.

864. Extract the square root of 25400544 in the senary scale, and of 32e75721 in the duodenary.

865. Which of the weights 1, 3, 32, &c. pounds must be selected to weigh 304 pounds?

866. If ab be an irreducible fraction, and b be any number except 9 or 3, show that when it is converted into a recurring decimal, the period will be divisible by 9, and the sum of the remainders a multiple of 6.

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867. If a+br+cr2+ +ka" represent a number of n + 1 digits; find the sum of all the numbers that can be formed by placing the digits in all possible positions.

868. There is a number consisting of 3 digits in geometrical progression; the number is to the sum of its digits=124: 7; and if 594 be added to it, the digits will be inverted: find the number.

869. If the number expressed by the last n digits of a number be divisible by 2", the number itself is divisible by 2".

870. Transform 8978 from a local value 11, and 3256 from a local value 7, to a system in which the local value is 12; and multiply the numbers together in this system.

871. Find the value of the circulating senary fraction 45.2534534, &c.

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