Sequence

In the arithmetic sequence is given:

S_n=2304, d=2, a_n=95

Calculate a₁ and n.

Correct answer:

a =  1
n =  48

Step-by-step explanation:

$$\begin{gathered} s=2304 \\ d=2 \\ b=95 \\ b=a+d\cdot (n-1) \\ s=n\cdot (a+b)\mathrm{/}2 \\ a=b-d\cdot (n-1) \\ 2s=n\cdot (a+b) \\ 2s=n\cdot (b-d\cdot (n-1)+b) \\ 2\ast 2304=n\ast (2\ast 95-2\ast (n-1)) \\ 2\cdot 2304=n\cdot (2\cdot 95-2\cdot (n-1)) \\ 2n^2-192n+4608=0 \\ a=2;b=-192;c=4608 \\ D=b^2-4ac=192^2-4\cdot 2\cdot 4608=0 \\ D=0 \\ {n}_{1,2}={\displaystyle \frac{-b\pm \sqrt{D}}{2a}}={\displaystyle \frac{192\pm \sqrt{0}}{4}}=48\pm \sqrt{0} \\ {n}_{1,2}=48\pm 0 \\ {n}_1=n_2=48 \\ \text{ Factored form of the equation: } \\ 2(n-48)(n-48)=0 \\ n>0 \\ n=n_1=48 \\ a=b-d\cdot (n-1)=95-2\cdot (48-1)=1 \end{gathered}$$

Our quadratic equation calculator calculates it.

$n=n_1=48=48$

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