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7. Write the third and fifth terms of an arithmetic sequence whose fourth term is 9 and the common difference is 2.

Sagot :

Sure, let's solve this step-by-step.

Given:
- The fourth term of the arithmetic sequence ([tex]\(a_4\)[/tex]) is 9.
- The common difference ([tex]\(d\)[/tex]) is 2.

We need to find the third term ([tex]\(a_3\)[/tex]) and the fifth term ([tex]\(a_5\)[/tex]).

Step 1: Finding the Third Term ([tex]\(a_3\)[/tex])

To find the third term of the sequence, we use the definition of an arithmetic sequence, where each term differs from the previous one by the common difference [tex]\(d\)[/tex].

The formula for the [tex]\(n\)[/tex]th term of an arithmetic sequence is given by:
[tex]\[ a_n = a_{n-1} + d \][/tex]

We can rearrange this formula to find the previous term:
[tex]\[ a_{n-1} = a_n - d \][/tex]

Substituting [tex]\(n = 4\)[/tex] for the fourth term:
[tex]\[ a_3 = a_4 - d \][/tex]

Given [tex]\(a_4 = 9\)[/tex] and [tex]\(d = 2\)[/tex]:
[tex]\[ a_3 = 9 - 2 \][/tex]
[tex]\[ a_3 = 7 \][/tex]

Step 2: Finding the Fifth Term ([tex]\(a_5\)[/tex])

Similarly, to find the fifth term of the sequence, we use the formula for the [tex]\(n\)[/tex]th term again:
[tex]\[ a_{n+1} = a_n + d \][/tex]

Substituting [tex]\(n = 4\)[/tex] for the fourth term:
[tex]\[ a_5 = a_4 + d \][/tex]

Given [tex]\(a_4 = 9\)[/tex] and [tex]\(d = 2\)[/tex]:
[tex]\[ a_5 = 9 + 2 \][/tex]
[tex]\[ a_5 = 11 \][/tex]

Conclusion:

The third term of the arithmetic sequence is [tex]\(7\)[/tex] and the fifth term is [tex]\(11\)[/tex].