Can someone explain how this question is done? - Thanks
In this question, you are given the kth term of a sequence. This means that if you plug in a value of k, you will get the term of the sequence numbered k. For example,
the first term, `P_1` will have k = 1 and `P_1 = 2^(1-1)/(1!) = 1/1 = 1` ,
the second term, `P_2` , will have k = 2 and `P_2 = 2^(2-1)/(2!) = 2^1/(1*2) = 1` ,
the third term, `P_3` , will have k = 3 and `P_3 = 2^(3-1)/(3!) = 2^2/(1*2*3) = 4/6=2/3` ,
and so on.
Note that the exclamation sign after k is called "factorial" and it means multiplying all integers starting with 1 and up to k:
`k! = 1*2*3*....(k-1)*k`
So, to find `P_(k+1)` , we need to plug in k+1 instead of k, that is, substitute k+1 for every k in the expression for `P_k` and then simplify:
`P_(k+1) = 2^(k+1 - 1)/((k+1)!) = 2^k/((k+1)!)`
This corresponds to the answer choice B.