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Find the sum of the first 21 terms of the arithmetic sequence, given π 41 plus π nine is equal to negative 232 and π 27 is equal to negative 130.
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In order to calculate the sum of the first π terms, we use the formula π of π is equal to π over two multiplied by two π plus π minus one multiplied by π.
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In this question, we want to find the sum of the first 21 terms.
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Therefore, π of 21 is equal to 21 divided by two multiplied by two π plus 20π.
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In order to answer the question, we therefore need to calculate the value of π and the value of π.
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π is the first term in the sequence and π is the common difference.
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The value of the πth term π π is equal to π plus π minus one multiplied by π.
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Weβre told that the 27th term π 27 is equal to negative 130.
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This means that π plus 26π is equal to negative 130.
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We will call this equation one.
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Weβre also told that the 41st term plus the ninth term is equal to negative 232.
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This means that π plus 40π plus π plus eight π is equal to negative 232.
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Simplifying this expression by grouping or collecting like terms gives us two π plus 48π is equal to negative 232.
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We can divide both sides of this equation by two.
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Two π divided by two is equal to π and 48 π divided by two is 24π.
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On the right-hand side, negative 232 divided by two is equal to negative 116.
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We will call this equation two.
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We now have a pair of simultaneous equations, which we can solve by elimination.
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When we subtract equation two from equation one, the πs cancel.
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26π minus 24π is equal to two π.
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Negative 130 minus negative 116 is the same as negative 130 plus 116.
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This is equal to negative 14.
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Dividing both sides of this equation by two gives us π is equal to negative seven.
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We can now substitute this value back in to equation one or equation two.
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Substituting into equation one gives us π plus 26 multiplied by negative seven is equal to negative 130.
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26 multiplied by negative seven is equal to negative 182.
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Simplifying this equation gives us π minus 182 equals negative 130.
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Adding 182 to both sides of this new equation gives us π is equal to 52.
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The first term of our arithmetic sequence is 52 and the common difference is negative seven.
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We can then substitute these values into our formula for π of 21.
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21 divided by two is equal to 10.5.
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Two multiplied by 52 is 104.
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20 multiplied by negative seven is negative 140.
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π of 21 is equal to 10.5 multiplied by 104 minus 140.
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104 minus 140 is equal to negative 36.
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Multiplying this by 10.5 gives us negative 378.
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The sum of the first 21 terms of the arithmetic sequence is negative 378.