Future Value of an Ordinary Annuity
where Sis the future value, Ris the payment at the end of
each period, iis the interest rate per period, and nis the
number of periods.
Sum of the Terms of an Infinite Geometric Sequence
with
S
a 1
1 r|r|< 1SRc11 i 2 n 1
id,If $5800 is deposited into an ordinary annuity at the end of each
quarter for 4 yr and interest is earned at 2.4% compounded quarterly,
thenandThe sum Sof the terms of an infinite geometric sequence with
and isS=1
1 -^12
=
1
1
2= 1 #
2
1
=2.
r=^12a 1 = 1S= 5800 c11 +0.006 216 - 1
0.006
d=$97,095.24.i= n= 4142 =16,0.024
4
R=$5800, =0.006,
CONCEPTS EXAMPLES
12.4 The Binomial Theorem
Factorials
For any positive integer n,
By definition,
Binomial Coefficient
General Binomial Expansion
For any positive integer n,
rth Term of the Binomial Expansion of
n!
1 r 12! 3 n 1 r 124!xn^1 r^12 yr^11 xy 2 n+yn.+
n!
1 n- 12 !1!+ xyn-^1n!
3! 1 n- 32!xn-^3 y^3 +Á+
n!
2! 1 n- 22!= xn+ xn-^2 y^2n!
1! 1 n- 12!xn-^1 y1 x+y 2 n
nCr r◊nn!
r! 1 nr 2!,
0!1.
n!n 1 n 121 n 22 Á 122112.
The eighth term of isSimplify.
=-15,360a^3 b^7. Multiply.= 1201 - 1282 a^3 b^7=
10 # 9 # 8
3 # 2 # 1a^31 - 227 b^710!
7!3!
a^31 - 2 b 271 a- 2 b 210= 16 x^4 - 96 x^3 + 216 x^2 - 216 x+ 81= 16 x^4 - 12182 x^3 + 54142 x^2 - 216 x+ 81= 24 x^4 - 41223 x^3132 + 61222 x^2192 - 412 x 21272 + 814!
3! 1!
12 x 21 - 323 + 1 - 324= 12 x 24 +4!
1! 3!
12 x 231 - 32 +4!
2! 2!
12 x 221 - 322 +12 x- 324=
5 # 4 # 3 # 2 # 1
3 # 2 # 1 # 2 # 15 C 3 = = 10
5!
3! 15 - 32!
=
5!
3!2!
4!= 4 # 3 # 2 # 1 = 24
708 CHAPTER 12 Sequences and Series
y=- 2 b, r= 8n=10, x=a,