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Answer:
[tex] k = 13[/tex]
Step-by-step explanation:
Let a3 be the first term (a1), then a5 will be the third term (a3).
Using Arithmetic Sequence Formula
[tex] a_{k} = a_{1} + (k - 1)d[/tex]
The following values are ...
[tex] a_{1} = 11 \\ a_{3} = 5 \\ k = 3 \\ d = ?[/tex]
Subtitute the given value
[tex] a_{k} = a_{1} + (k - 1)d \\ 7 = 11 + (3 - 1)d \\ 2d = 7 - 11 \\ 2d = - 4 \: (divide \: both \: sides \: by \: 2) \\ d = - 2[/tex]
Now that we found our common difference which is -2, let a5 be the first term and ak the last term. The following values are ...
[tex] a_{1} = 7 \\ a_{k} = - 9 \\ d = - 2 \\ k = \: ?[/tex]
Using Arithmetic Sequence Formula
[tex] a_{k} = a_{1} + (k - 1)d[/tex]
Substitute the given values
[tex] - 9 = 7 + (k - 1)(-2) \\ - 2k + 2 = - 9 - 7 \\ - 2k = - 9 - 7 - 2 \\ - 2k = - 18 \: (divide \: both \: sides \: by \: - 2) \\ k = 9[/tex]
So the value of k is 9 but since we adjusted earlier, we need to adjust the value of k
[tex] k = 9 + 4 \\ (we \: subtracted \: 4 \: to \: adjust \: a_{5} \: to \: a_{1}) \\ k = 13[/tex]
So the answer is 13