How to write an expression using sigma notations?

Question

If I have $4^{n-1}*6+4^{n-2}*6+4^{n-3}*6+...+4*6 + 6$ how do I rewrite such expression as sum notation? I do know the answer, I don't really understand how to go from this expression and make a sigma notation?

As I said, I know the answer, but I do not understand how to get there so I am asking more for an explanation rather than a plain answer — Martynas Krupskis, Sep 13 '19 at 13:16
Not sure what to tell you. Factor out the $6$. What's left is $(1+4+4^2+\dots +4^{n-1})$ which is literally $\sum_{i=0}^{n-1}4^i$, standard Geometric Series. — lulu, Sep 13 '19 at 13:18

score 1 · Accepted Answer · answered Sep 13 '19 at 13:13

1

You have terms of the form $4^i\times 6$, where $i$ starts from $n-1$ and finishes at $0$. In notation, $$\sum_{i=0}^{n-1} 4^i\times 6,$$ or, equivalently, $6 \sum_{i=0}^{n-1}4^i$.

answered Sep 13 '19 at 13:13

Luke Collins

8,748

Can we rewrite this as $6(4^n-1)$? – Martynas Krupskis Sep 13 '19 at 13:28
No, the sum evaluates to $6\times \tfrac13(4^n-1)$. – Luke Collins Sep 13 '19 at 13:33
Do you use sum of geometric sequence to figure that out? – Martynas Krupskis Sep 13 '19 at 14:27
Yes, the sum is $\frac{1-r^n}{1-r}=\frac{1-4^n}{1-4}$. – Luke Collins Sep 13 '19 at 14:31

How to write an expression using sigma notations?

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