Answer and step-by-step explanation:
*A recursive formula is a formula that defines each term of a sequence using preceding term(s). Recursive formulas must always state the initial term, or terms, of the sequence.
The given sequence is similar to a geometric sequence but there is no common ratio.
6 ÷ 2 = 3
-18 ÷ 6 = -3
-54 ÷ -18 = 3
*Observe that the quotients above are alternates of positive 3 and negative 3. Therefore, the hint is a negative ratio must have an exponent.
*A negative number with an exponent that will go up will have a product of an alternative of positive and negative. For example: (-5)^2 = 25, (-5)^3 = -125.
Then make a recursive formula using the first term and the hint above.
This is the recursive formula of the given sequence:
[tex] \\ a1 = 2 \\ an = a1( - 3) {}^{n - 1} \\ \\ where \: n≠2 or 3 [/tex]
a2 = 2(-3)^2-1
a2 = 2(-3)^1
a2 = 2(-3)
a2 = -6
-6 ≠ 6
a3 = 2(-3)^3-1
a3 = 2(-3)^2
a3 = 2(9)
a3 = 18
18 ≠-18
If find the second term or the third term of the given sequence, the answer will contradict the recursive formula.
Then use the recursive formula to find the next three terms which is the 5th, 6th and 7th terms of the given sequence.
a5 = 2(-3)^5-1
a5 = 2(-3)^4
a5 = 2(81)
a5 = 162
a6 = 2(-3)^6-1
a6 = 2(-3)^5
a6 = 2(-243)
a6 = -486
a7 = 2(-3)^7-1
a7 = 2(-3)^6
a7 = 2(729)
a7 = 1458
Therefore, the next three terms of the given sequence is 162, -486 and 1458.
