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I'm sorry if i wasn't so much expressive in the title because i have no idea how to synthesize this logic question.

here it is: A director wishes to know how many times her movie has been shown in a given theater. The theater's staff provides the following info: 1) at the first projection there's just 1 viewer; 2) at each projection, the number of viewers grows by 1 compared to the previous one; 3) 820 tickets are sold over all the projections. How many projections were held?

i have thought that $ y_{n+1}=y_n+1 $ and $ y_0 = 1 $ i know that $ y = 820 $ and i want to know $ n $.

can anyone help me, please. i hope that it wasn't wrong to write here. this is my first question..

1 Answers 1

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Ultimately the number of tickets sold in the first, second, third and so on till the nth session is $$1,2,\cdots,n $$ This is an Arithmetic Progression with starting term and common difference both equal to $1$.

Summing up the LHS using the formula $\frac {n (a_1+a _n)}{2} $, we get $$\frac {n (n+1)}{2}=820$$ This is a solvable quadratic in $n $. The answer is $\boxed {40 } $. Hope it helps.

On a side note, you can realise that we are just calculating the sum of the first $n $ natural numbers so directly use the relevant formula and get the answer.

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    thanks a lot, i am going to study your answer in order to better understand the question. thank you very much. should i change the title in something inherent with arithmetic Progression? – 2017-01-24
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    You can add the tag sequence and series. No problem, I will do it. – 2017-01-24