Determine whether each of the following is always, sometimes, or never true. If the answer is sometimes true, state the conditions under which it would be true.
sn+1 - sn = an+1
is the n and n+1 a subscript?
If so, then we appear to be talking about the sequence of partial sums of a sequence.
So, take a look...
sn = a1 + a1 + ... + an
s_n+1 = a1 + a2 + ... + a_n + a_n+1
clearly, s_n+1 = s_n + a_n+1
so, s_n+1 - s_n = a_n+1 always
To determine whether the equation sn+1 - sn = an+1 is always, sometimes, or never true, we need some additional information. Specifically, we need to know the relationship between sn, sn+1, and an+1.
Let's break down the possibilities:
1. If sn+1 is always equal to sn+an+1, then the equation sn+1 - sn = an+1 will always be true. This means that the difference between consecutive terms sn+1 and sn is exactly equal to an+1. In this case, the equation is always true.
2. If sn+1 is sometimes equal to sn+an+1, but not always, then the equation sn+1 - sn = an+1 will be sometimes true. This situation implies that there are certain conditions or patterns under which the equation holds. To determine the conditions, we would need to analyze the given sequence or mathematical relationship and identify any patterns or rules governing the values of sn, sn+1, and/or an+1.
3. If sn+1 is never equal to sn+an+1, then the equation sn+1 - sn = an+1 will never be true. This means that the difference between consecutive terms sn+1 and sn is never equal to an+1. In this case, the equation is never true for any values of the sequence.
To definitively assess whether the equation is always, sometimes, or never true, we would need more information about the specific sequence or relationship between the terms.