<s>[INST] Sum of the series : $1 + 2+ 4 + 7 + 11 +\cdots$ I got a question which says
$$ 1 + \frac {2}{7} + \frac{4}{7^2} + \frac{7}{7^3} + \frac{11}{7^4} + \cdots$$
I got the solution by dividing by $7$ and subtracting it from original sum. Repeated for two times.(Suggest me if any other better way of doing this). 
However now i am interested in understanding the series 1,2,4,7,11,..... In which the difference of the numbers are consecutive natural numbers. 
How to find the sum of 
$1+2+4+7+11+\cdots nterms$
This is my first question in MSE. If there are some guidelines i need to follow, which i am not, please let me know.
 [/INST] Let T = 1 + 2/7 + 4/7^2 + 7/7^3 + ... ----> (1)
Then 
  T/7 =     1/7 + 2/7^2 + 4/7^3 + ... ----> (2)
Now, 
(2)-(1) => 6T/7 = 1 + 1/7 + 2/7^2 + 3/7^3 ----> (3)
Hence, (1/7)(6T/7) =  1/7 + 1/7^2 + 2/7^3 ----> (4)
So, (3)-(4)=> (1 - 1/7)(6T/7) = 1 + 0 + 1/7^2 + 1/7^3 + ...
       => (6/7)(6T/7) = 1 + (1/7^2 + 1/7^3 + .... infinity)

       => 36T/49 = 1 + (1/7^2)/(1 - 1/7)    [Formula = a/(1-r)]

       => 36T/49 = 1 + (1/49)/(6/7)

       => T = (49/36)(1 + (1/49)(7/6))

       => T = 49/36 + (49/36)(1/49)(7/6)

       => T = 49/36 + 7/216

       => T = (294 + 7)/216

       => T = 301/216 

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