Question Video: Finding the Number of Terms That Make the Sum of a Given Arithmetic Sequence Negative Mathematics

Find the least number of terms needed to make the sum of the arithmetic sequence (15, 10, 5, ...) negative.

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Video Transcript

Find the least number of terms needed to make the sum of the arithmetic sequence 15, 10, five, and so on negative.

We recall that the sum of any arithmetic sequence, written 𝑆 sub 𝑛, is equal to 𝑛 over two multiplied by two 𝑎 plus 𝑛 minus one multiplied by 𝑑, where 𝑎 is the first term of the sequence and 𝑑 is the common difference. In this question, the first term of the sequence is 15. The numbers are decreasing by five. Therefore, 𝑑 is equal to negative five. Substituting these values into our formula, we have 𝑛 over two multiplied by two multiplied by 15 plus 𝑛 minus one multiplied by negative five.

We want this sum to be negative. Therefore, it must be less than zero. Two multiplied by 15 is equal to 30. By distributing the parentheses, we can multiply negative five by 𝑛 and negative five by negative one. This gives us negative five 𝑛 plus five. Simplifying the left-hand side of the inequality gives us 𝑛 over two multiplied by negative five 𝑛 plus 35 is less than zero. Multiplying both sides of this inequality by two gives us 𝑛 multiplied by negative five 𝑛 plus 35 must be less than zero.

When multiplying two numbers, if we want our answer to be negative or less than zero, one of the numbers must be negative. Since 𝑛 is the number of terms, this cannot be less than zero. In order for our inequality to be correct, negative five 𝑛 plus 35 must be less than zero. We can add five 𝑛 to both sides such that five 𝑛 is now greater than 35. Dividing both sides by five gives us 𝑛 is greater than seven.

In order for the sum of the arithmetic sequence to be negative, the number of terms 𝑛 must be greater than seven. We know that 𝑛 must be an integer or whole number value. Therefore, the least number of terms needed to make the sum negative is eight.

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