Evaluating Linear Expressions: Finding the Term-to-Term Rule for 2n+1 Sequence

Which expressions represent a term-to-term rule for the sequence shown below?

a. $5n+1-3n$

b. $2n+1$

c. $7n-1-5n$

d. $1-4n+6n$

To identify which expressions represent a term-to-term rule for a sequence, we'll simplify each expression:

• Expression a: $5n + 1 - 3n$
  Simplifying this, we combine like terms:
  $(5n - 3n) + 1 = 2n + 1$.
  This simplifies to a linear expression: $2n + 1$.
• Expression b: $2n + 1$
  This expression is already in its simplest linear form.
• Expression c: $7n - 1 - 5n$
  Simplifying this, we combine like terms:
  $(7n - 5n) - 1 = 2n - 1$.
  This simplifies to a linear expression: $2n - 1$.
• Expression d: $1 - 4n + 6n$
  Simplifying this, we combine like terms:
  $1 + (6n - 4n) = 1 + 2n$.
  This also simplifies to a linear expression: $2n + 1$.

After simplification:

• Expression a simplifies to $2n + 1$.
• Expression b is already $2n + 1$.
• Expression c simplifies to $2n - 1$, which is not identical to $2n + 1$ and does not match the sequence rule form identified in expressions a, b, and d.
• Expression d simplifies to $2n + 1$.

Thus, the expressions that represent a term-to-term rule of the form $2n + 1$ are

a, b, and d