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Arithmetic Progressions

What's the sum of the first 100 positive integers? How about the first 1000? Learn the fun and fast way to solve problems like this. See more

Level 3

         

\(a_1,a_2,a_3,\ldots, a_{98} \) are terms in an arithmetic progression with common difference 1 such that their sum is 137.

What is the sum of the even terms of this progression? That is, what is the value of \( a_2+a_4+a_6+\cdots+a_{98} \)?

1000 chocolates are placed on a table. On their covers, they are labelled \(1,2,3,4, \cdots ,1000\). Two friends Sandeep and Abhiram eat chocolates labelled in the arithmetic progressions \(3,6,9,12,15,\cdots\) and \(7,14,21,28,35,\cdots\) respectively. If ever there is any conflict between them i.e. if both are entitled to eat the same chocolate, Sandeep eats it. So, what is the total number of chocolates which are not eaten?

What is the largest positive integer that must be a divisor of the sum of an arithmetic sequence with 90 integer terms?


Try Part II

The interior angles of a convex polygon are in an arithmetic progression. If the smallest angle is \( 100^{\circ}\)and common difference is \(4^{\circ}\), then find the number of sides.

\[\large \frac ab \ , \ ab \ , \ a -b \ , \ a+b \]

Above shows real numbers that belong to an arithmetic progression in order. Find the next term of this sequence.

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