How to Find the First Term and the Common Difference?

To find the first term and common difference in an arithmetic sequence, you'll typically need at least two terms from the sequence. Let's explore how to do this, drawing on the reference provided. The video titled "FINDING FIRST TERM AND COMMON DIFFERENCE" provides some key insights.

Identifying Terms and Using the Formula

The video uses the notation a_m to denote a term in the sequence, where 'm' represents the position of that term.

• Example: If the second term is 24, it's noted as a_2 = 24.
• You typically need at least two different terms to work with, such as the second and fifth terms in this example.

Steps to Determine the First Term and Common Difference

1. Identify Two Terms:

   • You need at least two terms and their positions within the sequence. For example, let’s say you have the second term ( a_2) and the fifth term ( a_5).
   • From the video, a_2=24. Let's assume for the sake of explanation that a_5=33.

2. Use the Arithmetic Sequence Formula

• The general formula for an arithmetic sequence is a_n = a_1 + (n-1)d where:
  • a_n is the nth term.
  • a_1 is the first term.
  • d is the common difference.
  • n is the position of the term.

3. Set up a system of equations:

   • Using the identified terms and the arithmetic sequence formula:
     • For the second term: 24 = a_1 + (2-1)d which simplifies to 24 = a_1 + d (Equation 1)
     • For the fifth term: 33 = a_1 + (5-1)d which simplifies to 33 = a_1 + 4d (Equation 2)

4. Solve for d (Common Difference):

   • You can solve this system of equations using various methods such as substitution or elimination.
   • Using Elimination: Subtract equation 1 from equation 2:
     • 33 - 24 = (a_1 + 4d) - (a_1 + d)
     • 9 = 3d
     • d = 3

5. Solve for a_1 (First Term):

   • Now that you have the common difference (d=3), substitute this value back into either equation 1 or 2.
   • Using equation 1:
     • 24 = a_1 + 3
     • a_1 = 24 - 3
     • a_1 = 21

Example Summary:

• Given: a_2 = 24 , and (assuming for this example), a_5 = 33
• Calculated:
  • Common Difference: d = 3
  • First Term: a_1 = 21

Practical Insights

• More terms make it easier: Having more than two terms can make finding the common difference and first term easier, or you can use more terms for verification.
• Be careful with signs: Pay close attention to the signs when subtracting equations.
• Check your work: After finding the first term and the common difference, plug those values back into the sequence formula and verify they produce the given terms.