The sel IV of all 3 x 3 matrices A with Irace equal to zero is a subspace of M 33 . (See Exercise II in Section 4.3.) Determine a subset 5 of IV so Ihal span 5 = IV

ARTHMETIC AND GEOMETRIC SEQUENCES AND SERIES If a sequence follows a pattern of addition= ARITHMETIC SEQUENCE If a sequence follows a pattern of multiplication= GEOMETRIC SEQUENCE ARITHMETIC SEQUENCE - A list of numbers that follow a pattern of addition, - Ex. 4,7,10,13,16…. - Common difference (d)= the number that is added in order to create each term in the sequence. o To find d, subtract from each term that comes before it o In our example d= 3 - To find the n term of an arithmetic sequence use the equation - a na +1n-1)d a the nth term (the number in position n in the n= sequence) a 1=the first term in the sequence n= the position of the number in the sequence if the first time is not given to find the nth term, and you are given two terms use: a n a +bn-b)d GEOMETRIC SEQUENCE : A geometric sequence is a list of numbers that follows a pattern of multiplication And the number that multiplies each term in order to create the next term in the sequence is called common ratio, r (n-1) a n a *r 1 no first t:rm a =a *r (n-b) n b note: b has to be before n SUM OF ARITHEMETIC SERIES used to find the sum of the first n terms of an arithmetic series FOR SUM OF INFINITE GEO. SERIES. Only if -1