WEBVTT
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Find the least number of terms needed to make the sum of the arithmetic sequence 15, 10, five, and so on negative.
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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.
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In this question, the first term of the sequence is 15.
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The numbers are decreasing by five.
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Therefore, 𝑑 is equal to negative five.
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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.
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We want this sum to be negative.
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Therefore, it must be less than zero.
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Two multiplied by 15 is equal to 30.
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By distributing the parentheses, we can multiply negative five by 𝑛 and negative five by negative one.
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This gives us negative five 𝑛 plus five.
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Simplifying the left-hand side of the inequality gives us 𝑛 over two multiplied by negative five 𝑛 plus 35 is less than zero.
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Multiplying both sides of this inequality by two gives us 𝑛 multiplied by negative five 𝑛 plus 35 must be less than zero.
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When multiplying two numbers, if we want our answer to be negative or less than zero, one of the numbers must be negative.
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Since 𝑛 is the number of terms, this cannot be less than zero.
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In order for our inequality to be correct, negative five 𝑛 plus 35 must be less than zero.
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We can add five 𝑛 to both sides such that five 𝑛 is now greater than 35.
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Dividing both sides by five gives us 𝑛 is greater than seven.
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In order for the sum of the arithmetic sequence to be negative, the number of terms 𝑛 must be greater than seven.
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We know that 𝑛 must be an integer or whole number value.
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Therefore, the least number of terms needed to make the sum negative is eight.