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.
