Explanation:
an arithmetic series is a series of terms, where every term is built by using the previous term and adding or subtracting something specific.
this one is really simple :
the odd integers greater than 5 means we start with 7.
and we get the next odd integer by simply adding 2.
a1 = 7
a2 = a1 + 2 = 9
=>
an = an-1 + 2 = a1 + (n-1)×2
so, how many terms ?
99 is the last one.
an = 99 = a1 + (n-1)×2 = 7 + (n-1)×2
we need to solve for n
99 = 7 + (n-1)×2
92 = (n-1)×2
46 = n-1
n = 47
therefore, the series has 47 terms or elements.
the sum of all of them ?
the old trick :
99 + 7 = 106
97 + 9 = 106
95 + 11 = 106
...
so, we have 46/2 = 23 such pairs.
so, we get 23×106.
but is this enough ?
since we have 47 elements, there is one element left to be added.
which one ?
when we build these pairs we reduce the left side by 2 22 times (as we have 23 pairs, and the first one is the starting point). and we add to the right side 2 22 times. so + and - 44 on either side.
that gives us
99-44 = 55
7+44 = 51
therefore, the one missing element is the one in the middle : 53.
so, the whole sum is
23×106 + 53 = 2438 + 53 = 2491