This frequently gives students problems; therefore let’s think about it this manner. Suppose k = 10. Then Sk would be S10, the sum of first 10 terms and ak+1 would be a11, the 11th term in the sequence. S10 + a11 would be the sum of 10 terms plus the 11th term that would be the sum of first 11 terms.
The other way to write ‘for each and every positive integer n’ is for each and every positive integer n. This works since Z is the set of integers, therefore Z+ is the set of positive integers. The upside down A is symbol for ‘for all’ or ‘for every’ or ‘for each’ and the symbol which looks similar to a weird e is the ‘element of’ symbol. Therefore technically, the statement is stating ‘for each and every n which is an element of the positive integers’, however it is simpler to state ‘for each and every positive integer n’.
Therefore, we can see that the left hand side equivalents the right hand side for the first term, therefore we have established the primary condition of mathematical induction.
On left hand side, Sk + ak+1 signifies the ‘sum of the first k terms’ plus ‘the k+1 term’, that gives us the sum of first k+1 terms, Sk+1.