Which of the following represents the general term for the sequence 1, 3, 5, 7, 9, . . .?
2n - 1, n, n + 2, 2n

Solution:

It is given that

The sequence that is given to us is 1, 3, 5, 7, 9, ...

Let us assume a₁ = 1, a₂ = 3, a₃ = 5, a₄ = 7, a₅ = 9

Then,

a₂ - a₁  = 3 - 1 = 2

a₃ - a₂ = 5 - 3 = 2

a₄ - a₃ = 7 - 5 = 2

a₅ - a₄ = 9 - 7 = 2.

Hence, from the above simplification we can see that the common difference is 2.

So, the nth term of an arithmetic progression whose first term a₁ = 1 and common difference d = 2 is given by,

a_n = a₁ + (n - 1) d

Substituting the values:

a_n = 1 + (n - 1) 2

= 1 + 2n - 2

= 2n - 1

Therefore, the general term for the sequence 1, 3, 5, 7, 9, . . . is 2n - 1.

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Which of the following represents the general term for the sequence 1, 3, 5, 7, 9, . . .?
2n - 1, n, n + 2, 2n

Summary:

The general term for the sequence 1, 3, 5, 7, 9, . . . is 2n - 1.