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Write the explicit formula for exch sequenu below

Write the explicit formula for exch sequenu below

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Question

Write the explicit formula for exch sequenu below.
1.) 0,6,16,30,62,0,6,16,30,62,cdots

Best Answer

The explicit formula for the given sequence is:
an=n2+6a_(n)=n^(2)+6, where a1=0a_(1)=0 and nZ+n inZ^(+).
To find the explicit formula, we need to determine the pattern in the sequence.
The difference between consecutive terms is:
60=66-0=6,
166=1016-6=10,
3016=1430-16=14,
6230=3262-30=32,
We observe that the differences are not constant. Therefore, the sequence is not arithmetic.
The next step is to calculate the differences between the differences:
106=410-6=4,
1410=414-10=4,
3214=1832-14=18,
We can see that the second differences are constant. This tells us that the sequence can be modeled by a quadratic equation of the form an=An2+Bn+Ca_(n)=An^(2)+Bn+C.
We can use the first three terms of the sequence to determine the values of AA, BB, and CC.
When n=1n=1, a1=0a_(1)=0;
0=A(1)2+B(1)+C0=A(1)^(2)+B(1)+C,
A+B+C=0A+B+C=0
When n=2n=2, a2=6a_(2)=6;
6=A(2)2+B(2)+C6=A(2)^(2)+B(2)+C,
4A+2B+C=64A+2B+C=6
When n=3n=3, a3=16a_(3)=16;
16=A(3)2+B(3)+C16=A(3)^(2)+B(3)+C,
9A+3B+C=169A+3B+C=16
We have three equations with three unknowns. Solving for AA, BB, and CC, we get:
A=1A=1,
B=0B=0,
C=6C=6
Therefore, the explicit formula for the sequence is an=n2+6a_(n)=n^(2)+6, where a1=0a_(1)=0 and nZ+n inZ^(+).
To verify this formula, we can check the remaining terms in the sequence:
a2=22+6=10a_(2)=2^(2)+6=10
a3=32+6=15a_(3)=3^(2)+6=15
a4=42+6=22a_(4)=4^(2)+6=22
a5=52+6=31a_(5)=5^(2)+6=31
a6=62+6=42a_(6)=6^(2)+6=42
This confirms that the explicit formula an=n2+6a_(n)=n^(2)+6 generates the sequence 0,6,16,30,62,0,6,16,30,62,cdots.