This section is restricted to the expansion of (1 +x)n and to the various tech-niques arising from such expansions. The techtech-niques are based on the Pascal triangle and its basic properties, but the proofs of these properties will be left until Section 5B.
Some Expansions of (1 +x)n: Here are the expansions of (1 +x)n for low values ofn.
The calculations have been carried out using two rows so that like terms can be written above each other in columns. In this way, the process by which the coefficients build up can be followed better.
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174 CHAPTER5: The Binomial Theorem CAMBRIDGEMATHEMATICS3 UNITYEAR12r
(1 +x)0 = 1 (1 +x)1 = 1 +x
(1 +x)2 = 1(1 +x) + x(1 + x)
= 1 + x + x + x2
= 1 + 2x + x2
(1 +x)3 = 1(1 +x)2+x(1 + x)2
= 1 + 2x + x2 + x + 2x2+x3
= 1 + 3x + 3x2+x3 (1 +x)4 = 1(1 +x)3+x(1 + x)3
= 1 + 3x + 3x2+ x3 + x + 3x2+ 3x3+x4
= 1 + 4x + 6x2+ 4x3+x4
Notice how the expansion of (1+x)2 has 3 terms, that of (1+x)3 has 4 terms, and so on. In general, the expansion of (1 +x)n has n + 1 terms, from the constant term inx0= 1 to the term inxn. Be careful — this is inclusive counting — there aren + 1 numbers from 0 to n inclusive.
The Pascal Triangle and the Addition Property: When the coefficients in the expansions of (1+x)n are arranged in a table, the result is known as the Pascal triangle. The table below contains the first five rows of the triangle, copied from the expansions above, plus the next four rows, obtained by continuing these calculations up to (1 +x)8.
Coefficient of:
n x0 x1 x2 x3 x4 x5 x6 x7 x8
0 1
1 1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
5 1 5 10 10 5 1
6 1 6 15 20 15 6 1
7 1 7 21 35 35 21 7 1
8 1 8 28 56 70 56 28 8 1
Four properties of this triangle should quickly become obvious. They will be used in this section, and proven formally in the next.
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BASIC PROPERTIES OF THE PASCAL TRIANGLE: 1. Each row starts and ends with 1.
2. Each row is reversible.
3. The sum of each row is 2n.
4. [The addition property] Every number in the triangle, apart from the 1s, is the sum of the number directly above, and the number above and to the left.
The first three properties should be reasonably obvious after looking at the ex-pansions at the start of the section. The fourth property, called the addition property, however, needs attention. Three numbers in the Pascal triangle above have been boxed as an example of this — notice that
1 + 3 = 4.
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CHAPTER5: The Binomial Theorem 5A The Pascal Triangle 175r
The expansions on the first page of this chapter were written with the columns aligned to make this property stand out. For example, 1 + 3 = 4 arises like this
— in the expansion of (1 +x)4, the coefficient ofx3 is the sum of the coefficients ofx3 and x2 in the expansion of (1 +x)3.
The whole Pascal triangle can be constructed using these rules, and the first question in the following exercise asks for the first thirteen rows to be calculated.
Using Pascal’s Triangle: The following worked exercises illustrate various calculations involving the coefficients of (1 +x)n for low values ofn.
WORKEDEXERCISE: Use the Pascal triangle to write out the expansions of:
(a) (1− x)4 (b) (1 + 2a)6 (c) (1− 23x)5 SOLUTION:
(a) (1− x)4 = 1 + 4(−x) + 6(−x)2+ 4(−x)3+ (−x)4
= 1− 4x + 6x2− 4x3+x4
(b) (1 + 2a)6 = 1 + 6(2a) + 15(2a)2+ 20(2a)3+ 15(2a)4+ 6(2a)5+ (2a)6
= 1 + 12a + 60a2+ 160a3+ 240a4+ 192a5+ 64a6
(c) (1−23x)5 = 1 + 5(−23x) + 10(−23x)2+ 10(−23x)3+ 5(−23x)4+ (−23x)5
= 1−103 x +409 x2−8027x3+8081x4−24332 x5 WORKEDEXERCISE:
(a) Write out the expansion of
1 + 5
x
2
, then write out the first four terms in the expansion of (1− x)8.
(b) Hence find, in the expansion of
1 +5
x
2
(1− x)8:
(i) the term independent ofx, (ii) the term inx.
SOLUTION: (a)
1 +5
x
2
= 1 + 10x−1+ 25x−2
(1− x)8 = 1− 8x + 28x2− 56x3+· · · (b) Hence in the expansion of
1 + 5
x
2
(1− x)8:
(i) constant term = 1× 1 + (10x−1)× (−8x) + (25x−2)× (28x2)
= 1− 80 + 700
= 621.
(ii) term inx = 1 × (−8x) + (10x−1)× (28x2) + (25x−2)× (−56x3)
=−8x + 280x − 1400x
=−1128x.
WORKEDEXERCISE: By expanding the first few terms of (1 + 0·02)8, find an ap-proximation of 1·028 correct to five decimal places.
SOLUTION:
(1 + 0·02)8 = 1 + 8× 0·02 + 28 × (0·02)2 + 56× (0·02)3 + 70× (0·02)4 + · · ·
= 1 + 0·16 + 0·0112 + 0·000 448 + 0·000 011 20 + · · ·
=.. 1·171 66
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176 CHAPTER5: The Binomial Theorem CAMBRIDGEMATHEMATICS3 UNITYEAR12r
WORKEDEXERCISE: Find the value ofk if, in the expansion of (1 + 2kx)6: (a) the terms in x4 and x3 have coefficients in the ratio 2 : 3,
(b) the terms in x2,x3 and x4 have coefficients in arithmetic progression.
SOLUTION: (1 + 2kx)6 =· · · + 15(2kx)2+ 20(2kx)3+ 15(2kx)4+· · · prominent place for use in the rest of this chapter.
2. Using Pascal’s triangle of binomial coefficients, give the expansions of each of the following:
(a) (1 +x)6
3. Continue the calculations of the expansions of (1 +x)n at the beginning of this section, expanding (1 +x)5 and (1 +x)6 in the same manner. Keep your work in columns, so that the addition property of the Pascal triangle is clear.
4. Find the specified term in each of the following expansions.
(a) For (1 +x)11: (i) find the term in x2, (ii) find the term inx8.
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CHAPTER5: The Binomial Theorem 5A The Pascal Triangle 177r
5. Sketch on one set of axes:(a) y = (1 − x)0, y = (1 − x)2, y = (1 − x)4, y = (1 − x)6. (b) y = (1 − x)1, y = (1 − x)3, y = (1 − x)5.
6. Expand (1 +x)9 and (1 +x)10, and show that the sum of the coefficients of the second expansion is twice the sum of the coefficients in the first expansion.
D E V E L O P M E N T
7. Without expanding, simplify:
(a) 1 + 3(x − 1) + 3(x − 1)2+ (x − 1)3
(b) 1− 6(x + 1) + 15(x + 1)2− 20(x + 1)3+ 15(x + 1)4− 6(x + 1)5+ (x + 1)6 8. Find the coefficient ofx4 in the expansion of (1− x)4+ (1− x)5+ (1− x)6. 9. Find integersa and b such that:
(a) (1 +√ 11. Verify by direct expansion, and by taking out the common factor, that:
(a) (1 +x)4− (1 + x)3 =x(1 + x)3 (b) (1 +x)7− (1 + x)6 =x(1 + x)6
12. (a) Expand the first few terms of (1 +x)6, hence evaluate 1·0036 to five decimal places.
(b) Similarly, expand (1− 4x)5, and hence evaluate 0·965 to five decimal places.
(c) Expand (1 +x)8− (1 − x)8, and hence evaluate 1·0028− 0·9988 to five decimal places. 15. Determine the value of the term independent ofx in the expansion of:
(a) (1 + 2x)4
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178 CHAPTER5: The Binomial Theorem CAMBRIDGEMATHEMATICS3 UNITYEAR12r
16. (a) In the expansion of (1 +x)6:
(i) find the term inx2, (ii) find the term inx3, (iii) find the ratio of the term inx2 to the term in x3,
(iv) find the values of (i), (ii) and (iii) whenx = 3.
(b) In the expansion of
1 + 2
3x
7
:
(i) find the term inx−5, (ii) find the term inx−6, (iii) find the ratio of the term inx−5 to the term inx−6,
(iv) find the values of (i), (ii) and (iii) when x = 2.
17. (a) When (1 + 2x)5 is expanded in increasing powers ofx, the third and fourth terms in the expansion are equal. Find the value ofx.
(b) When (1 +x)5, where x = 0, is expanded in increasing powers of x, the first, second and fourth terms in the expansion form a geometric sequence. Find the value of x.
(c) When (1+x)7 is expanded in increasing powers ofx, the fifth, sixth and seventh terms in the expansion form an arithmetic sequence. Find the value ofx.
18. (a) Find the coefficients of x4 and x5 in the expansion of (1 +kx)8. Hence find k if these coefficients are in the ratio 1 : 4.
(b) Find the coefficients of x3 and x4 in the expansion of (1 +kx)6. Hence find k if these coefficients are in the ratio 8 : 3.
(c) Find the coefficients ofx5 andx6 in the expansion of (1−34kx)9. Hence findk if these coefficients are equal.
19. Use Pascal’s triangle to help evaluate the integrals arising from the following questions.
(a) Find the area bounded by the curvey = x(1 − x)5 and thex-axis, where 0 ≤ x ≤ 1.
(b) Find the area bounded by the curvey = x4(1− x)4 and thex-axis, where 0 ≤ x ≤ 1.
(c) Find the volume of the solid formed when the region between thex-axis and the curve y =
x(1 − x)3, for 0≤ x ≤ 1, is revolved around the x-axis.
20. If $P is invested at the compound interest rate R per annum for n years, and interest is compounded annually, the accumulated amount is $A, where A = P (1 + R)n.
(a) Write down as decimals all terms in the expansion of (1 + 0·04)3.
(b) Hence find the amount to which an investment of $1000 will grow, if it is invested for 3 years at a rate of 4% per annum, and interest is compounded annually.
21. By writing (1 +x + 3x2)6 as (1 +A)6, where A = x + 3x2, expand (1 +x + 3x2)6 as far as the term inx3. Hence evaluate (1·0103)6 to four decimal places.
22. [Patterns in Pascal’s triangle] Check the following results using the triangle you con-structed in question 1. (These will not be proven until later.)
(a) The sum of the numbers in the row beginning 1,n, . . . is equal to 2n.
(b) If the second member of a row is a prime number, all the numbers in that row excluding the 1s are divisible by it.
(c) [The hockey stick pattern] Starting at any 1 on the left side of the triangle, go diagonally downwards any number of steps. Then the sum of these numbers is the number directly below the last number. For example, if you start at the 1 on the left hand side of the row 1, 3, 3, 1 and move down the diagonal 1, 4, 10, 20 the total of these numbers, namely 35, is found directly below 20.
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CHAPTER5: The Binomial Theorem 5B Further Work with the Pascal Triangle 179r
(d) [The powers of 11] If a row is made into a single number by using each element as a digit of the number, the number is a power of 11 (except that after the row 1, 4, 6, 4, 1, the pattern gets confused by carrying).
(e) Find the diagonal and the column containing the triangular numbers, and show that adding adjacent pairs gives the square numbers.
23. [These geometrical results should be related to the numbers in the Pascal triangle.]
(a) Place three points on the circumference of a circle. How many line segments and triangles can be formed using these three points?
(b) Place four points on the circumference of a circle. How many segments, triangles and quadrilaterals can be formed using these four points?
(c) What happens if five points are placed on the circle.
(d) How many pentagons could you form if you placed seven points on the circumference of a circle?
E X T E N S I O N
24. [The Pascal pyramid] By considering the expansion of (1 +x + y)n, where 0 ≤ n ≤ 4, calculate the first five layers of the Pascal pyramid.