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Find the arithmetic sequence in which the sum of the first and third terms equals negative 142, and the sum of its third and fourth terms equals negative 151.
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We denote the first term of any arithmetic sequence by the letter π.
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The common difference is denoted by the letter π.
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This is the difference between each of the terms in the sequence.
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The second term of the sequence is, therefore, equal to π plus π.
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Adding π to this gives us the third term π plus two π.
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This pattern continues, giving us an πth term formula of π plus π minus one multiplied by π.
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We are told in this question that the sum of the first and third terms is negative 142.
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This means that π plus π plus two π is equal to negative 142.
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Simplifying this gives us two π plus two π equals negative 142.
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We could divide both sides of this equation by two.
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However, weβll work out the second equation in this question first.
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The third and fourth terms have a sum of negative 151.
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This means that π plus two π plus π plus three π is equal to negative 151.
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Grouping or collecting the like terms gives us two π plus five π is equal to negative 151.
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We now have two simultaneous equations that we can solve to calculate the values of π and π.
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When we subtract equation one from equation two, the πβs cancel.
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Five π minus two π is equal to three π.
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Subtracting negative 142 is the same as adding 142 to negative 151.
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This is equal to negative nine.
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Dividing both sides of this equation by three gives us π is equal to negative three.
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We now need to substitute this value into equation one or equation two to calculate π.
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Substituting into equation one gives us two π plus two multiplied by negative three is equal to negative 142.
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This simplifies to two π minus six is equal to negative 142.
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Adding six to both sides gives us two π is equal to negative 136.
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And finally, dividing by two gives us π equals negative 68.
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As the first term of the sequence is negative 68 and the common difference is negative three, then the arithmetic sequence is negative 68, negative 71, negative 74, and so on.