The theorem for finding the sum of n terms of an Arithmetic Sequence is:
Sn = n/2[ 2a + (n - 1)d ] = n/2(a + an)
However, I am having a problem with the proof of this which is given in the book. I will show the proof given and stop at the point where I'm having a problem understanding.
Sn = a1 + a2 + a3 + ... + an ;...........................sum of first n terms
= a + (a + d) + (a + 2d) + ... + [a + (n - 1)d] ;....uses an = a + (n - 1)d
= (a + a + ... + a) + [d + 2d + ... + (n - 1)d] ;.....rearranging terms
= na + d[1 + 2 + ... + (n - 1)] ;.........................there are n a terms and factors out d
= na + d[ (n - 1)n / 2 ] ;..................................inside the brackets using property 1 + 2 + 3 + ... + n = n(n+1)/2
It is at this point in the proof that I have a problem, since n(n-1)/2 is not the same as n(n+1)/2.