omvamen71

2022-10-08

Write an equation for the nth term of the geometric sequence −2,10,−50,...

ralharn

Dividing a term with the previous one gives the result −5. This means that to get the next term, we multiply by −5. The sequence also starts at −2, which is quite unconventional to express in terms of −5. Therefore, since the problem doesn't restrict us in any way, we can say that the equation is
$-2\cdot {\left(-5\right)}^{n-1}$
This is because, like I mentioned above, to get to the next term we need to multiply by −5. Therefore, to get to the nth term, we need to multiply by ${\left(-5\right)}^{n-1}$.
As for why the exponent is n−1 and not just n, this is because for n=1, we are talking about the first term, which doesn't have anything to do with −5. Because of this, it's convenient to "cover" the first term with our equation by utilizing the fact that
${a}^{0}=1$ for any real nonzero a.
In other words, in order for the equation to include the case of the first number −2 (when n=1) we need an expression that multiplies −2 with not n, but n−1 instances of −5, otherwise the first term would be 10 (and that's our second term, and so on).

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