If you were given the three equations
And then the values of are asked for some , then you can get the answer by long expansions, squaring, getting needed terms etc, but there is a smart way too !
Some days ago, i had shared a set by name Bashing Unavailable...Awesome problems , which was on problems of this type , for .
If you haven't tried the set, I prefer telling you to try it before you read this note.
You can get the thing for by algebraic manipulations (and also for further values of ) , for example this "part of the solution" which I knowingly wrote to the first problem of the set.
This way, you can get the values.... but here goes the smarter way, the . My way is similar to my friend Aamir Faisal Ansari,a worth following person's way...
See that as shown in the image, we can get values of and
Next, let's define a sequence as
Then, we do the following algebraic manipulation which will give us the generalisation
(something = the extra terms that will come in the expansion of first term of the RHS)
For the extra terms, we have
Thus, we have
From this we get the recurrence relation
And because we know the values of all the coefficients in this thing, we get the recurrence relation
From this recurrence relation, , you can get the value of all the further terms like a cakewalk !
Reshare if this is helpful, and try to solve the last part of it, the part of the set - Bashing Unavailable Part 6