Answer for the box: -5n+2
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Work Shown:
[tex]a_1 = -3 = \text{ first term}\\\\d = -5 = \text{ common difference}\\\\a_n = a_1 + d(n-1) = \text{ nth term}\\\\a_n = -3 + (-5)(n-1)\\\\a_n = -3 -5n + 5\\\\a_n = -5n + 2\\\\[/tex]
The common difference is due to us adding -5 to each term to get the next one. Example: -3+(-5) = -8, or -8+(-5) = -13.
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As a way to check if we have the correct nth term formula, plug in values of n to see if we get the correct sequence value
For instance, if n = 1, then
[tex]a_n = -5n+2\\\\a_1 = -5(1)+2\\\\a_1 = -5+2\\\\a_1 = -3\\\\[/tex]
Which matches with the first term given to us.
Or if n = 2, then,
[tex]a_n = -5n+2\\\\a_2 = -5(2)+2\\\\a_2 = -10+2\\\\a_2 = -8\\\\[/tex]
That matches the second term given to us. I'll let you check the others.
