I'm currently doing some numeric sequence problems on an Algebra worksheet. But I've stumbled upon this problem and I don't seem to be able to solve it.
The problem is as follows:
The general term of [imath]\left ( a_{n} \right )[/imath] is [imath]a_{n} = \frac{1}{\left ( n+1 \right )^{2}}[/imath] .
Find the first five terms of [imath]\left ( b_{n} \right )[/imath] , the general term of which is
[math]b_{n} = \left ( a_{n} \right )^{2}-a_{n-1}[/math] .
So far I've managed to find the first 6 terms of [imath]\left ( a_{n} \right )[/imath] :
[math]a_{1} = \frac{1}{4} ; a_{2} = \frac{1}{9} ; a_{3} = \frac{1}{16} ; a_{4} = \frac{1}{25} ; a_{5} = \frac{1}{36} ; a_{6} = \frac{1}{49} ;[/math] .
But I don't know how to go about finding [imath]\left ( b_{1} \right )[/imath] because I would need [imath]a_{n-1}[/imath] which is [imath]\left ( a_{0} \right )[/imath]. I've thought about finding [imath]\left ( a_{0} \right )[/imath] and then substituting it in [imath]\left ( b_{1} \right )[/imath] but I don't know if that is the right approach.
The problem is as follows:
The general term of [imath]\left ( a_{n} \right )[/imath] is [imath]a_{n} = \frac{1}{\left ( n+1 \right )^{2}}[/imath] .
Find the first five terms of [imath]\left ( b_{n} \right )[/imath] , the general term of which is
[math]b_{n} = \left ( a_{n} \right )^{2}-a_{n-1}[/math] .
So far I've managed to find the first 6 terms of [imath]\left ( a_{n} \right )[/imath] :
[math]a_{1} = \frac{1}{4} ; a_{2} = \frac{1}{9} ; a_{3} = \frac{1}{16} ; a_{4} = \frac{1}{25} ; a_{5} = \frac{1}{36} ; a_{6} = \frac{1}{49} ;[/math] .
But I don't know how to go about finding [imath]\left ( b_{1} \right )[/imath] because I would need [imath]a_{n-1}[/imath] which is [imath]\left ( a_{0} \right )[/imath]. I've thought about finding [imath]\left ( a_{0} \right )[/imath] and then substituting it in [imath]\left ( b_{1} \right )[/imath] but I don't know if that is the right approach.