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Sagot :
Given:
The third term of an arithmetic progression is,
[tex]a_3=13[/tex]The thirteenth term of the arithmetic progression is,
[tex]a_{13}=43[/tex]The objective is to find the 17th term of the sequence.
Explanation:
The general formula for the nth term of an arithmetic sequence is,
[tex]a_n=a+(n-1)d\text{ . . . . . . (1)}[/tex]The expression for the third term can be written as,
[tex]\begin{gathered} a_3=a+(3-1)d \\ 13=a+2d\text{ . . . . (2)} \end{gathered}[/tex]Similarly, the expression for the thirteenth term can be written as,
[tex]\begin{gathered} a_{13}=a+(13-1)d \\ 43=a+12d \\ a=43-12d\text{ . . . . . .(3)} \end{gathered}[/tex]To find d:
Now, substitute equation (3) in equation (2).
[tex]\begin{gathered} 13=(43-12d)+2d \\ 13-43=-10d \\ -30=10d \\ d=\frac{30}{10} \\ d=3 \end{gathered}[/tex]To find a :
Now, substitute the value of d in equation (3).
[tex]\begin{gathered} a=43-12(3) \\ a=43-36 \\ a=7 \end{gathered}[/tex]Thus, the value of a is 7 and the value of d is 3.
To find a17:
Now, the 17th term can be calculated from equation (1) as,
[tex]a_{17}=7+(17-1)3[/tex]On further solving the above equation,
[tex]\begin{gathered} a_{17}=7+16(3) \\ =7+48 \\ =55 \end{gathered}[/tex]Hence, the value of a17 of the arithmetic progression is 55.value of a17 of thear
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