Question Video: Finding the General Term of an Arithmetic Sequence given the Values of Two Terms

Find, in terms of 𝑛, the general term of an arithmetic sequence whose ninth term is −717 and sixteenth term is −1347.

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Video Transcript

Find in terms of 𝑛 the general term of an arithmetic sequence whose ninth term is negative 717 and sixteenth term is negative 1347.

The general term of any arithmetic sequence with first term 𝑎 and common difference 𝑑 is given by 𝑇𝑛 is equal to 𝑎 plus 𝑛 minus one multiplied by 𝑑. In this example, the ninth term is equal to negative 717. This means that 𝑎 plus eight 𝑑 is equal to negative 717. We were also told that the sixteenth term is equal to negative 1347. This means that 𝑎 plus 15𝑑 is equal to negative 1347.

We now have a pair of simultaneous equations which we can solve to work out the value of 𝑎 and the value of 𝑑. Subtracting equation one from equation two gives us seven 𝑑 is equal to negative 630 as 15𝑑 minus eight 𝑑 is seven 𝑑 and negative 1347 minus negative 717 is equal to negative 630. Dividing both sides of this equation by seven gives us a value for 𝑑 of negative 90. The common difference in the arithmetic series is negative 90.

Substituting this value of 𝑑 back into equation one gives us 𝑎 plus eight multiplied by negative 90 is equal to negative 717. Eight multiplied by negative 90 is negative 720. Adding 720 to both sides of this equation gives us a value for 𝑎, the first term of the arithmetic sequence, equal to three.

This means that an arithmetic sequence with ninth term negative 717 and sixteenth term negative 1347 has a first term 𝑎 equal to three and a common difference equal to negative 90. Substituting these values into the equation 𝑇 of 𝑛 equals 𝑎 plus 𝑛 minus one multiplied by 𝑑 gives us three minus 90 multiplied by 𝑛 minus one. Expanding or multiplying out the parenthesis or bracket gives us negative 90𝑛 plus 90. This gives us the general term negative 90𝑛 plus 93.

An arithmetic sequence whose ninth term is negative 717 and sixteenth term is negative 1347 has a general term negative 90𝑛 plus 93.

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