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Staircase of quote increments

Mental mathMedium

A spread widens by $3$ , then $7$ , then $11$ , and so on, each increment $4$ larger than the last, ending with an increment of $43$ . What is the total widening?

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Hint

Count the terms first, then use the average of the ends.

Solution

The increments are $3, 7, 11, \dots, 43$ , with common difference $4$ .
The count is $(43 - 3) \div 4 + 1 = 11$ .
The average of the first and last terms is $(3 + 43) \div 2 = 23$ .
The total is $11 \times 23 = 253$ .

Answer: $253$

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