Chapter 11 Math Review

Pascal's Triangle

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1

Finding the Partial Sum of a Geometric Sequence

Find the sum of the first five terms of the geometric sequence: 1,0.7,0.49,0.343... Since a=1, r=0.7 and n=5, we can use our formula and find that the sum is 2.7731

Finding a Particular Term in Binomial Expansion

Find term that contains x^5 in the expansion of (2x+y)^20 20!/15!(20-15)! (2x)^5 * y^15 = 20!/15!*5! 32x^5 * y^15 =496,128 x^5y^15

Finding particular Term coefficient in Binomial Expansion

Find the coefficient of x^8 in expansion of (x^2+ 1/x )^10 (x^2)^6 * 1/x4 ^=x^8 r=6 10!/ 6!(10-6)! =210

Finding the Terms of a Geometric Sequence

Find the eight term of the geometric sequence 5,15,45... In order to find r, we find the ratio of any two consecutive terms, so 45/15=3. Then, because a=5 and r=3, we can use our formula to find that the eight term is 10935

Finding the terms of a recursively defined sequence

Find the first five terms of the sequence defined recursively by a₁=1 and an=3(an-₁+2) ANS: 1,9,33,105,321

Finding Terms of an Arithmetic Sequence

Find the first six term and the 300th term of the arithmetic sequence 13, 7,... 7-13= -6, so d=-6 Therefore, an=13-6(n-1) First six terms are 13,7,1,-5,-11,-17 300th term is -1781

Finding the Partial Sum of an Arithmetic Sequence

Find the sum of the first 40 terms of the arithmetic sequence : 3,7,11,15... a₁=3, d=4 Therefore, the sum is 3240

Different Partial Sum of Arithmetic Sequence Problem

Find the sum of the first 50 odd numbers. The odd numbers form an arithmetic sequence with a=1 and d=2, thus the nth term of the sequence is an=1+2(n-1), and therefore, the 50th term would be 99. Then, using the arithmetic sum formula, we find our answer of 2500

Mathematical Induction

For each natural number n let P(n) be a statement depending on n. Suppose that the following 2 conditions are satisfied. 1. P(1) is true 2. For every natural number k ; if P(k) is true then P(k+1) is true. Then P(n) is true for all natural numbers n.

Sequence

Function f whose domain is the set of natural numbers

Finding the Number of Terms in a Partial Sum

How many of terms of the arithmetic sequences 5,7,9,... must be added to get 572? Since a=5 and d=2, we can substitute 572 into our sum of an arithmetic sequence equation and find that n= 22 or -26. Since it asks for the number of terms, however, the only logical answer is 22 terms.

Index of summation (Summation Variable)

Is the "K" in the sigma notation n ∑ak=a1+a2+a3...+an k=1

Partial Sum

S1=a1 S2=a1+a2 S3=a1+a2+a3 ... Sn=a1+a2+a3...+an

Find the sum of infinite geometric series 2+2/5+2/25+2/125+........+2/5^n+......

Since |r|<1 so S=2(/1- 1/5) = 5/2

Sum of Arithmetic series

Sn= (an-a1)/2 Sn=n(a1-an/2)

Sum of Infinite Geometric Series

Sn= a/1-r |r|<1

Partial Sum of Geometric Sequence

Sn=a(1-r^n/1-r)

Terms

Values in a sequence

Arithmetic Series

an= a1+d(n-1)

Geometric Sequence

an= ar^n-1

Binomial

expression of the form a+b

Summation Notation

n ∑ak=a1+a2+a3...+an k=1

binomial coefficient

n!/ r!(n-r)!

Combination

n!/r!(n-r)! A collection of things in which the order does not matter

General Term that contains A^r in expansion of (a+b)

n!/r!(n-r)! a^r b^n-r

n! is n factorial

n!= 1*2*3*4*......*(n-1)*n

Recursive

the nth term of the sequence depends on all or some of the terms preceding it