Question

Write the first five terms of a sequence. Don’t make your sequence too simple. Write both an explicit formula and a recursive formula for a general term in the sequence. Explain in detail how you found both formulas.

Answer 1

Answer: The answer is 5, 10, 20, 40 and 80.

Step-by-step explanation: We are to write the first five terms of a sequence, along with the explicit and recursive formula for the general term of the sequence.

Let the first five terms of a sequence be 5, 10, 20, 40 and 80. These terms are taken from a geometric sequence with first term
`a_1 5` and common ratio
`r=2.`

Therefore, we have

`a_2=a_1* r,\\\\a_3=a_2* r,\ldots`

Therefore, the recursive formula is

`a_(n+1)=2a_n,~~a_1=5.`

And explicit formula is

`a_n=a_1r^(n-1).`

Answer 2

Let the first five terms of a sequence be, 4,8,12,20,24

Ok, let me explain the meaning of term explicit and Recursive formula.

Explicit formula ,is the general formula to find any term of the sequence.

And, Recursive formula, is the method by which, we can find the nth term of the sequence if (n-1)th term of the sequence is known.

So,the explicit formula for the sequence is,

y = 4 n, where , n is any natural number.

And the recursive formula is .

y = 4 (n-1), with the help of (n-1)th term we can find nth term.

First term =4=4 × 1

Second term =8 =4 × 2

Third term =12=4×3

Fourth term =16 =4 × 4

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(n-1) th term =4 × (n-1)

nth term =4 × n

Applying the simple procedure by looking at the first , second and the way the next term goes , the general and recursive formula is obtained.

As, you said you don't want simpler sequence

Consider the first five terms of the sequence, 0, 3,8,15, 24.

Explicit formula =n² + 2 n

Recursive formula=(n-1)²+2(n-1)

If,you will look at the sequence, it is neither Arithmetic nor geometric .

First term =0=0²+0×0

Second term =3=1+2=1²+2 × 1

Third term =8=4+4=2²+2×2

Fourth term =15=9+6=3²+2×3

Fifth term =24=16 +8=4²+2×4

So, Explicit formula= n²+ 2 n, where n is a whole number.