Video: Finding the Sum of a Given Number of Terms of an Arithmetic Sequence Under a Given Condition

Find the sum of the first 21 terms of the arithmetic sequence, given π‘Ž_(41) + π‘Ž_(9) = βˆ’232 and π‘Ž_(27) = βˆ’130.

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Video Transcript

Find the sum of the first 21 terms of the arithmetic sequence, given π‘Ž 41 plus π‘Ž nine is equal to negative 232 and π‘Ž 27 is equal to negative 130.

In order to calculate the sum of the first 𝑛 terms, we use the formula 𝑆 of 𝑛 is equal to 𝑛 over two multiplied by two π‘Ž plus 𝑛 minus one multiplied by 𝑑. In this question, we want to find the sum of the first 21 terms. Therefore, 𝑆 of 21 is equal to 21 divided by two multiplied by two π‘Ž plus 20𝑑. In order to answer the question, we therefore need to calculate the value of π‘Ž and the value of 𝑑.

π‘Ž is the first term in the sequence and 𝑑 is the common difference. The value of the 𝑛th term π‘Ž 𝑛 is equal to π‘Ž plus 𝑛 minus one multiplied by 𝑑. We’re told that the 27th term π‘Ž 27 is equal to negative 130. This means that π‘Ž plus 26𝑑 is equal to negative 130. We will call this equation one. We’re also told that the 41st term plus the ninth term is equal to negative 232. This means that π‘Ž plus 40𝑑 plus π‘Ž plus eight 𝑑 is equal to negative 232.

Simplifying this expression by grouping or collecting like terms gives us two π‘Ž plus 48𝑑 is equal to negative 232. We can divide both sides of this equation by two. Two π‘Ž divided by two is equal to π‘Ž and 48 𝑑 divided by two is 24𝑑. On the right-hand side, negative 232 divided by two is equal to negative 116. We will call this equation two. We now have a pair of simultaneous equations, which we can solve by elimination.

When we subtract equation two from equation one, the π‘Žs cancel. 26𝑑 minus 24𝑑 is equal to two 𝑑. Negative 130 minus negative 116 is the same as negative 130 plus 116. This is equal to negative 14. Dividing both sides of this equation by two gives us 𝑑 is equal to negative seven. We can now substitute this value back in to equation one or equation two. Substituting into equation one gives us π‘Ž plus 26 multiplied by negative seven is equal to negative 130. 26 multiplied by negative seven is equal to negative 182. Simplifying this equation gives us π‘Ž minus 182 equals negative 130. Adding 182 to both sides of this new equation gives us π‘Ž is equal to 52.

The first term of our arithmetic sequence is 52 and the common difference is negative seven. We can then substitute these values into our formula for 𝑆 of 21. 21 divided by two is equal to 10.5. Two multiplied by 52 is 104. 20 multiplied by negative seven is negative 140. 𝑆 of 21 is equal to 10.5 multiplied by 104 minus 140. 104 minus 140 is equal to negative 36. Multiplying this by 10.5 gives us negative 378.

The sum of the first 21 terms of the arithmetic sequence is negative 378.

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