Exercise 3: Quadratic sequences

Transcription

1 Exerise 3: s Problem 1: Determine whether eah of the following sequenes is: a linear sequene; a quadrati sequene; or neither ;17;3;53;80; 3 p ;6 p ;9 p ;1 p ;15 p ; 1;,5;5;8,5;13; ;6;10;14;18; 5;19;41;71;109; 3;9;16;1;7; k;8k;18k;3k;50k; ;6;10 ;16; ; Answer 1: = 9; 15; 1; 7 = 6. First differene: = 3p Linear sequene 3. = 1,5;,5; 3,5; 4,5 = 1 1/6

2 4. First differene: = 4 Linear sequene 5. = 14; ; 30; 38 = 8 6. Neither 7. = 6k; 10k; 14k; 18k = 4k 8. = 3,5; 4,5; 5,5; 6,5 = 1 Problem : A quadrati pattern is given by. Find the values of and if the sequene starts with the following terms: Answer : Starting with the first term, we have and T1 : For the seond term, we use and T : T n = n + bn + b 1 ; ; 7 ; 14 ; n = 1 T 1 = (1) + b(1) + ( 1) = 1 + b + = b + n = = = 1 /6

3 T = () + b() + () = 4 + b + = b + Now we must solve these equations simultaneously. We an do this by substitution, but here we will show the solution using the 'elmination' method (whih means subtrating one equation from the other to anel the 's). = b + ( = b + ) 0 = b Finally, alulate the value of. As usual for simultaneous equations, this means that we must b = 0 = b + substitute the into either of the equations we used above. Let's use the equation. b = 0 = b + = (0) + = b = 0 The final answers are and. NOTE: Now we know that the general term of the sequene is. We an use this to hek our answers. We know that. Substitute into the general formula to hek: T n T3 = T 3 = 7 n = 3 = n = (3) = (9) + 0 = 7 T n = n Problem 3: a ; a ; 3 a ; 5 a ; are the first 4 terms of a sequene. Is the sequene linear or quadrati? Motivate your answer.. What is the next term in the sequene? 3. Calulate T 100. Answer 3: = = 3/6

4 T T 1 = a a = a T 3 T = 3a ( a ) = a T4 T3 = 5a ( 3 a ) = a This is an arithmeti sequene sine there is a ommon differene of terms. a between onseutive. T5 = 5 a + ( a ) = 7a 3. T n T 100 T 100 = a + (n 1)d = a + (99)( a ) = a 198a = 197a Problem 4: T n = n + bn + b Given, determine the values of and if the sequene starts with the terms: ; 7 ; 14 ; 3 ; Answer 4: Starting with the first term, we have and T1 : For the seond term, we use and T : Now we must solve these equations simultaneously. We an do this by substitution, but here we will show the solution using the 'elimination' method (whih means subtrating one equation from the other to anel the 's). n = 1 = T 1 = (1) + b(1) + () = 1 + b + 1 = b + n = = 7 T = () + b() + (7) = 4 + b + 3 = b + 4/6

5 Finally, alulate the value of. As usual for simultaneous equations, this means that we must b = substitute the into either of the equations we used above. Let's use the equation. The final answers are 3 = b + (1 = b + ) = b b = 1 = b + 1 = () + 1 = 1 = b + b = = 1 and. T n = n + n 1 T 3 = 14 n = 3 NOTE: Now we know that the general term of the sequene is. We an use this to hek our answers. We know that. Substitute into the general formula to hek: T n n = + n 1 T 3 = (3) + (3) 1 = (9) = 14 Problem 5: The first term of a quadrati sequene is 4, the third term is 34 and the ommon seond differene is 10. Determine the first six terms in the sequene. Answer 5: Let T T T 1 And T 3 T Seond differene 4; 14; 34; 64; 104; 154 = x = x 4 = 34 x = ( ) ( ) T3 T T T1 = (34 x) (x 4) 10 = 38 x x = 8 x = /6

6 Problem 6: A quadrati sequene has a seond term equal to 1, a third term equal to 6 and a fourth term equal to 14. Determine the seond differene for this sequene.. Hene, or otherwise, alulate the first term of the pattern. Answer 6:. T3 T T 4 T 3 Seond differene T 1 = = 7 = 6 (1) = 7 = 14 ( 6) = 8 = 1 6/6