Which term of the AP 150,147,144.....is it’s first negative term?

1.	Which term of the AP 150,147,144.....is it’s first negative term?
Answer» GIVEN : Airthemtic progression series 150,147,144,..... To FIND : First negative term in the series THEORY : ${\purple{\boxed{\large{\bold{a _{n} = a + (n - 1)d}}}}}$ Theory : Given Ap series: 150,147,144,... In this series ; First term ,a = 150 common DIFFERENCE ,d = 147-150=-3 we have to find first negative term in the given AP series . ⇒ The value of a+(n-1) d < 0 $\sf\:a+(n-1)d$ $\sf\:150+(n-1)(-3)$ $\sf150-3n +3$ $\sf\:153-3n$ $\sf3n>153$ $\sf\:n >\frac{153}{3}$ $\sf\:n>51$ As n is a integer, the first value which satisfies the above condition is n =52 Now, $\sf\:a _{52} =a + (52- 1)d$ $\sf\:a _{52} =150+ (52- 1)(-3)$ $\sf\:a _{52} =150-156+3$ $\sf\:a _{52} =-3$ Therefore first negative no is -3 $\rule{200}2$ More About Arithmetic Progression: 1) Genral term of an Ap $\sf\:a_n=a+(n-1)d$ 2)Sum of n terms of an AP given by : $\sf \: S_{n} = \dfrac{1}{2}(2a+ (n - 1)d)$

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