Thursday 19 September 2019

sequences and series - Arithmetic progression , find $n$



Given arithmetic progresion $20,17,14,\ldots$



Find smallest value of $n$ so $y_n<0$.



I can find value of $n$ by mind , but do not know how to write the solution.


Answer



Assuming $y_0=20$, your arithmetic progression has closed form




$$y_n=20-3n$$



Now let us solve the following inequality.



$$0>20-3n$$
$$3n>20$$
$$n>\frac{20}{3}=7-\frac{1}{3}$$



The minimum integer value of $n$ that satisfies the condition is

$n=7$.


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