Consider the following sequence:
\n0, 6, 24, 60, 120, 210
\nWhich one of the following numbers will come next in the sequence ?<\/p>\n
[amp_mcq option1=”240″ option2=”290″ option3=”336″ option4=”504″ correct=”option3″]<\/p>\n
Now, find the differences between these first differences (second differences):
\n18 – 6 = 12
\n36 – 18 = 18
\n60 – 36 = 24
\n90 – 60 = 30<\/p>\n
Now, find the differences between these second differences (third differences):
\n18 – 12 = 6
\n24 – 18 = 6
\n30 – 24 = 6<\/p>\n
Since the third differences are constant (6), the sequence is based on a cubic polynomial. A common pattern for such sequences is $n^3 – c \\cdot n$ or similar. Let’s try $n^3 – n$:
\nFor n=1: $1^3 – 1 = 1 – 1 = 0$ (Matches the first term)
\nFor n=2: $2^3 – 2 = 8 – 2 = 6$ (Matches the second term)
\nFor n=3: $3^3 – 3 = 27 – 3 = 24$ (Matches the third term)
\nFor n=4: $4^3 – 4 = 64 – 4 = 60$ (Matches the fourth term)
\nFor n=5: $5^3 – 5 = 125 – 5 = 120$ (Matches the fifth term)
\nFor n=6: $6^3 – 6 = 216 – 6 = 210$ (Matches the sixth term)<\/p>\n