Answer:
n = 12
Step-by-step explanation:
After you divide by 9, you have ...
[tex]\displaystyle\sum\limits^n_{m=1}m=78[/tex]
The sum of numbers 1..n is given by ...
[tex]\displaystyle\sum\limits^n_{m=1}m=\dfrac{n(n+1)}{2}=78[/tex]
This resolves to the quadratic ...
n² +n -156 = 0
(n +1/2)² -156.25 = 0 . . . . . complete the square
n = -0.5 + √156.25 = -0.5 + 12.5
n = 12
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Comment on the sum
The sum of numbers 1 to n can be treated as the sum of the arithmetic series whose first term is 1 and whose last term is n. The average term is (1+n)/2, and the sum is the product of that and the number of terms:
n(n+1)/2 = sum of numbers 1 to n.
