Thinking@TheCulture is a series of questions that we, as tutors feel that are useful in helping students think and improve their understanding.
Thinking Math@TheCulture is curated by KS. More of him can be found here
This is a standard summation question. I’m interested in the last part only.
The answer to (ii) is written there by the student. I’ll only do the solution to (iv).
From (ii), we know that
So we start with (as given in (iv) of the question),
Next, we replace with . This is to make the term in the summation similar, disregarding the lower and upper limit.
*Observe the lower limits and upper limits should change. Since lower limit is while upper limit is .
Next, we try to make the lower limit the same as in (ii), that is appear.
*The second term is when . It’s to balance the equation out mathematically.
Lastly, we can sub the formula from (ii) in by replacing by directly. It will affect our RHS.