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This problem trains for: SAT-I, AMC-8, GMAT.

A start-up company processes mail orders. In the first day, it processes only 5 orders. Due to successful advertisement, the number of processed orders increases, on average, by 3 orders every day. How many orders did the company process in its 7-th week of existence?

The number of orders is an arithmetic sequence with first term 5 and common difference 3.

Let us sum the orders over the first 6 weeks and then over the first 7 weeks and then subtract, to find only the orders processed in the 7-th week alone. 6 weeks have a total of 42 days and 7 weeks total 49 days.

For n days, the sum of orders is:

where there are only n-1 increases over n days (stands to reason).

Here, we notice a sum which we can replace with its formula (but not applying it blindly, since we have only n-1 numbers to sum up, while the formula is for n):

Now we substitute the number of days in the formula to find out the total number of orders processed during the period:

The difference represents the number of orders processed in the 7-th week alone: