Proof of Sum of Geometric Series Formula (using proof by induction) 

I was on my deathbed before this video but you have enlightened me with the knowledge required to success

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I have questions can you please answer it please Use mathematical induction to prove this formula for the sum of a finite number of terms of a geometric progression with initial term \( a \) and common ratio \( r \)\[ \sum_{j=0}^{n} a r^{j}=a+a r+a r^{2}+\cdots+a r^{n}=\frac{a r^{n+1}-a}{r-1} \]when \( r eq 1 \)where \( n \) is a non-negative integer.

Is this the only sequence and series proof you need to know?