a×(5+c)=a×6+a×c is
called
Answers
A,B,C,D, … can be thought of as a sequence of n terms, where the first term, n=1 corresponds to A. Likewise, n=2 for the second term which is B, and so on.
A,B,C, D …
2, 3, 5, 8, …
A solution for the nth term (any term in the sequence) can be obtained. As the terms do not appear to differ by the same number, it can be a bit tricky to deduce by observation. However, notice that A and B differ by 1, B and C differ by 2, C and D differ by 3, etc. And looking the differences 1,2,3, they differ by 1, so this sequence has two “layers”, which tells us the formula for the nth term will be quadratic.
We can use the general formula an^2+bn+c= d, where a,b, and c are constants we want to find, n is a variable for the number of the term corresponding to each letter, and d is a variable for the value of each term/letter.
As there are three constants, we need three equations.
Plugging in d=2 when n=1, d=3 when n=2, and d=5 when n=3, we have
a +b+c=2
4a+2b+c=3
9a+3b+c=5
Solving these simultaneously using a graphing calculator or algebraically will yield a=1/2, b=-1/2, c=2, so our general equation becomes
1/2n^2–1/2n+2=d
Since we want to find F, and F corresponds to n=6, we can substitute in n=6 and get d=17.
Note: This method may seem unnecessary for this question, but it is useful when the term you are finding is far off, e.g finding the term when n=25. Hope it helps!