Loan 5

Abdul takes a loan of 200000 from Ali and agrees to repay in several installments. Each installment begins with the 2nd, exceeding the previous one by 1000. If the first installment is 500, find how many installments will be necessary to be wiped out of the loan. Completely

Correct answer:

n =  20

Step-by-step explanation:

$$\begin{gathered} a=200000 \\ {a}_1=500 \\ d=1000 \\ a=\frac{a_1+a_n}{2}n \\ a=\frac{a_1+a_1+(n-1)\cdot d}{2}\cdot n \\ 200000\cdot 2=(500\cdot 2+(n-1)\cdot 1000)\cdot n \\ 200000\cdot 2=(500\cdot 2+(n-1)\cdot 1000)\cdot n \\ -1000n^2+400000=0 \\ 1000n^2-400000=0 \\ 1000=2^3\cdot 5^3 \\ 0 \\ 400000=2^7\cdot 5^5 \\ \text{GCD}(1000,0,400000)=2^3\cdot 5^3=1000 \\ {n}^2-400=0 \\ {n}_{1,2}=\pm \sqrt{400}=\pm 20 \\ {n}_1=20 \\ {n}_2=-20 \\ n=n_1=20 \end{gathered}$$

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