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If (a_1, a_2..... a_17) satisfy

1 + a_2 + a_3 = 1, 
a_2 + a_3 + a_4 = 3, 
a_3 + a_4 + a_5 = 5, 

...

a_{15} + a_{16} + a_{17} = 29,
a_{16} + a_{17} + a_{1} = 31, 
a_{17} + a_{1} + a_{2} = 33,

what is the value of a_17?

 Oct 25, 2020
 #1
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Your arithmetic sequence:

 

1 - The first term is: - 1/3

 

2 - The common difference is: 2/3

 

     -1/3 + 1/3 + 1 = 1

       1/3 + 1 + 5/3 =3

       1 + 5/3 + 7/3=5

       .

       . 

17th term =First term + (Common difference * Number of term - 1)

                  = -1/3 + (2/3 * (17 - 1))

                  = -1/3 + (2/3 * 16)

                  = -1/3 +    10 2/3

                  =  10 + 1/3

                  = 10 1/3 - which is the 17th term

 Oct 26, 2020
edited by Guest  Oct 26, 2020
 #2
avatar+25598 
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If (a_1, a_2..... a_17) satisfy

1 + a_2 + a_3 = 1,
a_2 + a_3 + a_4 = 3,
a_3 + a_4 + a_5 = 5,

...

a_{15} + a_{16} + a_{17} = 29,
a_{16} + a_{17} + a_{1} = 31,
a_{17} + a_{1} + a_{2} = 33,

what is the value of a_17?

 

\(\begin{array}{|lrcll|} \hline (1) & 1 + a_2 + a_3 &=& 1 \\ (2) & a_2 + a_3 + a_4 &=& 3 \\ (3) & a_3 + a_4 + a_5 &=& 5 \\ (4) & a_4 + a_5 + a_6 &=& 7 \\ (5) & a_5 + a_6 + a_7 &=& 9 \\ \ldots \\ (14) &a_{14} + a_{15} + a_{16} &=& 27 \\ (15) &a_{15} + a_{16} + a_{17} &=& 29 \\ (16) &a_{16} + a_{17} + a_{1} &=& 31 \\ (17) &a_{17} + a_{1} + a_{2} &=& 33 \\ \hline \end{array} \begin{array}{|rcll|} \hline (1) & 1 + a_2 + a_3 &=& 1 \\ & \mathbf{a_2 + a_3} &=& \mathbf{0} \ \text{or}\ \mathbf{a_3 =-a_2} \\ \hline (2) & a_2 + a_3 + a_4 &=& 3 \\ & 0 + a_4 &=& 3 \\ & \mathbf{a_4} &=& \mathbf{3} \\ \hline \end{array}\)

 

\(\begin{array}{|l|rcll|} \hline (3)-(2) & a_5 &=& 2 + a_2 \\ (4)-(3) & a_6 &=& 2 + a_3 \\ (5)-(4) & a_7 &=& 2 + a_4 \\ (6)-(5) & a_8 &=& 2 + a_5 \\ (7)-(6) & a_9 &=& 2 + a_6 \\ (8)-(7) & a_{10} &=& 2 + a_7 \\ (9)-(8) & a_{11} &=& 2 + a_8 \\ (10)-(9) & a_{12} &=& 2 + a_9 \\ (11)-(10) & a_{13} &=& 2 + a_{10} \\ (12)-(11) & a_{14} &=& 2 + a_{11} \\ (13)-(12) & a_{15} &=& 2 + a_{12} \\ (14)-(13) & a_{16} &=& 2 + a_{13} \\ (15)-(14) & a_{17} &=& 2 + a_{14} \\ (16)-(15) & a_1 &=& 2 + a_{15} \\ (17)-(16) & a_2 &=& 2 + a_{16} \\ \hline \end{array}\)

 

\(\begin{array}{|rcll|} \hline a_3 &=& -a_2 \\ a_4 &=& 3 \\ \hline a_5 &=& 2 + a_2 \\ a_6 = 2 + a_3 &=& 2 - a_2 \\ a_7 = 2 + a_4 &=& 2 + 3 = 5 \\ \hline a_8 = 2 + a_5 &=& 2 + 2 + a_2 = 4+a_2\\ a_9 = 2 + a_6 &=& 2+2 - a_2 = 4 - a_2 \\ a_{10} = 2 + a_7 &=& 2 + 5 = 7 \\ \hline a_{11} = 2 + a_8 &=& 2 + 4 + a_2 = 6 + a_2\\ a_{12} = 2 + a_9 &=& 2 + 4 - a_2 = 6 - a_2 \\ a_{13} = 2 + a_{10} &=& 2 + 7 = 9 \\ \hline a_{14} = 2 + a_{11} &=& 2 + 6 + a_2 = 8 + a_2\\ a_{15} = 2 + a_{12} &=& 2 + 6 - a_2 = 8 - a_2 \\ a_{16} = 2 + a_{13} &=& 2 + 9 = 11 \\ \hline a_{17} = 2 + a_{14} &=& 2 + 8 + a_2 = 10 + a_2\\ a_{1} = 2 + a_{15} &=& 2 + 8 - a_2 = 10 - a_2 \\ \mathbf{a_2} = 2 + a_{16} &=& 2 + 11 \mathbf{= 13} \\ \hline a_{17} &=& 10 + a_2 \quad | \quad a_2 = 13 \\ a_{17} &=& 10 + 13 \\ \mathbf{a_{17}} &=& \mathbf{23} \\ \hline \end{array}\)

 

... so ...

\(\begin{array}{|rcll|} \hline a_1 &=& -3 \\ \hline a_2 &=& 13 \\ a_3 &=& - 13 \\ a_4 &=& 3 \\ a_5 &=& 15 \\ a_6 &=&-11 \\ a_7 &=& 5 \\ a_8 &=& 17 \\ a_9 &=& -9 \\ a_{10} &=& 7 \\ a_{11} &=& 19 \\ a_{12} &=& -7 \\ a_{13} &=& 9 \\ a_{14} &=& 21 \\ a_{15} &=& -5 \\ a_{16} &=& 11 \\ a_{17} &=& 23 \\ a_1 &=& -3 \\ a_2 &=& 13 \\ \hline \text{check} && a_{17} + a_1 + a_2 = 23-3+13 = 33\ \checkmark \\ \hline \end{array}\)

 

 

laugh

 Oct 28, 2020

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