3. Use the binomial theorem to find the coefficient of x° in (x+2)3 2. Use the L theorem to find the coeficient of x6 y3 in (2.
3. Use the binomial theorem to find the coefficient of x° in (x+2)3 2. Use the L theorem to find the coeficient of x6 y3 in (2.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:For now we will be content to accept the binomial theorem wi
2. Use the binomial theorem to find the coefficient of x° y° in (x + y)13
3. Use the binomial theorem to find the coefficient of x³ in (x+2)13
4. JUse the binomial theorem to find the coefficient of xº y³ in (3x– 2y)ª.
+y)=x?+ 7x®y+21x®y² + 35x*y³ +35x®y* +212 +1z° +j_
6. Use Definition 3.2 (page 85) and Fact 1.3 (page 13) to show E- = 2"
10. Show that the formula k(") = n(j) is true for all integers n,k with 0sks
n.
The Ineluston-2
37 The Inel
Many countin
of two finite s
First we
must equal
A and then
get IAI+B
counted ea
For another example,
(2a – b)*
%3D
= ((2a)+(-b))*
+ 4(2a)°(-b)+ 6(2a)(-b)² +4(2a)aha
16a –32a®b+24a²b² – 8ab³ + b4
%3D
Therefor
LAI+B|
Exercises for Section 3.6
Fa
if
1. Write out Row 11 of Pascal's triangle.
No
so Fa
to th
5. Use the binomial theorem to show Eo4) = 2".
form
excl
7. Use the binomial theorem to show E-o 3* (") = 4".
8. Use Fact 3.5 (page 87) to derive Equation 3.3 (page 90).
9. Use the binomial theorem to show (6)–(i)+6)-(5)+(4) – - -(-1)" (2) =0, 6
in fa
Cor
Ex
ma
11. Use the binomial theorem to show 9" = E"-,(-1)* () 10"-k.
12, Show that ()() = OC)-
13. Show that () = 6) + C) + () + () +
14. The first five rows of Pascal's triangle appear in the digits of powers of 11:11-
11= 11, 112 121, 11 = 1331 and 114 = 14641, Why is this so? Why doeste
pattern not continue with 11?
th
%3D
(i
+...+
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