Question Video: Finding the Sum of an Arithmetic Sequence Given the General Term Mathematics

Find the sum of the first 10 terms of the sequence 𝑎_𝑛, where 𝑎_𝑛 = 2𝑛 + 4.

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Video Transcript

Find the sum of the first 10 terms of the sequence 𝑎 sub 𝑛, where 𝑎 sub 𝑛 is equal to two 𝑛 plus four.

There are a few ways of approaching this problem. One way would be to calculate the first and last terms of the sequence. As there are 10 terms, these are denoted by 𝑎 sub one and 𝑎 sub 10. The first term will be equal to two multiplied by one plus four. This is equal to six. The tenth term 𝑎 sub 10 is equal to two multiplied by 10 plus four. This is equal to 24. We can now use the formula 𝑆 sub 𝑛 is equal to 𝑛 over two multiplied by 𝑎 plus 𝑙, where 𝑎 is equal to six, the first term, and 𝑙 is equal to 24, the 10th or last term. 𝑆 sub 10 is equal to 10 over two multiplied by six plus 24. This simplifies to five multiplied by 30, giving us an answer for the sum of the first 10 terms of the sequence of 150.

An alternative method would be to have recognized our sequence is linear. Therefore, the common difference 𝑑 is equal to two, the coefficient of 𝑛. If 𝑎 sub 𝑛 is equal to two 𝑛 plus four, our sequence is six, eight, 10, and so on. We could then use the formula that 𝑆 sub 𝑛 is equal to 𝑛 over two multiplied by two 𝑎 plus 𝑛 minus one multiplied by 𝑑. Substituting in our values here gives us 𝑆 sub 10 is equal to 10 over two multiplied by two multiplied by six plus 10 minus one multiplied by two. This simplifies to five multiplied by 12 plus 18, which once again is equal to five multiplied by 30, which gives us an answer of 150.

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