( ; a 3= 2 3 = 1;

(Example2)Determine the rst six terms of the dened sequences, and give their associated series.

(1). f2 ng(2). f1 + 2 n+ 3 n2g(3).

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f( 1) ng(4). f1 + 2 + 3 + 4 + +ngS olutions : We denote the nth term of a sequence by an, andS= a1 +a2 +a3 +a4 +a5 +a6:(1). an =f2 ngFirst six terms:a 1 = 21 = 1 ; a2= 22 = 0 ; a3= 23 = 1; a4= 24 = 2; a5= 25 = 3; a6= 26 = 4Associated series: S= a1 +a2 +a3 +a4 +a5 +a6 = 1 + 01 2 3 4 = 9(2). an =f1 + 2 n+ 3 n2gFirst six terms: a1 = 6; a2= 17; a3= 34; a4= 57; a5= 86; a6= 121Associated series: S= a1 +a2 +a3 +a4 +a5 +a6 = 6 + 17 + 34 + 57 + 86 + 121 = 321(3).

an =f( 1) ngFirst six terms: a1 =1; a2= 1; a3=1; a4= 1; a5=1; a6= 1Associated series: S= a1 +a2 +a3 +a4 +a5 +a6 =1 + 1 1 + 1 1 + 1 = 0(4). an =f1 + 2 + 3 + 4 + +ngFirst six terms: a1 = 1; a2= 3; a3= 6; a4= 10; a5= 15; a6= 21Associated series: S= a1 +a2 +a3 +a4 +a5 +a6 = 1 + 3 + 6 + 10 + 15 + 21 = 56(1). How many terms are there in an arithmetic sequence with rst term 1, common dier-ence -3, and last term -41?S olution :a1 = 1; d=3; an=41 :Find n.an = 1 + (n 1)( 3) = 411(n 1) = 41 1 3 = 14)n= 152

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