Write this as sum: $a_{0}+4a_{1}+7a_{2}+10a_{3}+13a_{4}$

Question

Write this as sum: $a_{0}+4a_{1}+7a_{2}+10a_{3}+13a_{4}$

(Hint: This is an arithmetic sequence)

From hint (arithmetic), I know that the trick of this sequence is that the difference of 2 neighbor numbers is always $3$.

But really no idea how to write this as sum?

$$\sum_{i=0}^{4}something \cdot a_{i}$$

I don't know how to make this difference $3$ in this sum very complicated? : /

Answer 1

I think you are searching for

$$\sum_{i=0}^4 (3i+1)a_i.$$

Edit

How to find this ?

You were on the right track when you said that the difference between to consecutive terms is $3$ (without the $a_i$). You then know that the factor of the arithmetic-geometric progression is $3$. Then you just have to find the "$+1$".