-4 is the first negative term. The progression is 24,20,16,12,8,4,0,-4,...
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It is an Arithmetic Progression with a constant difference of 11 and first term 15.
In an arithmetic progression the difference between each term (except the first) and the one before is a constant. In a geometric progression, their ratio is a constant. That is, Arithmetic progression U(n) - U(n-1) = d, where d, the common difference, is a constant and n = 2, 3, 4, ... Equivalently, U(n) = U(n-1) + d = U(1) + (n-1)*d Geometric progression U(n) / U(n-1) = r, where r, the common ratio is a non-zero constant and n = 2, 3, 4, ... Equivalently, U(n) = U(n-1)*r = U(1)*r^(n-1).
There are 5 common differences between seventh and twelfth terms, so the CD is 2.5/5 ie 0.5. First term is therefore 15 - 6 x 0.5 = 12.
If a is the first term and r the common difference, then the nth term is tn = a * (n-1)r So t16 = a + 15r Then 6*t16 = 6(a + 15r) or 6a + 90r No further simplifiaction is possible.
An arithmetic sequence is usually given by a formula in which the nth term, T(n), is given in terms of the first term, a, and the common difference, d: t(n) = a + d*(n-1) where n= 1, 2, 3, etc An alternative is to define it iteratively. Thus: t1 = a tn = tn-1 + d , where n = 2, 3, 4, etc