Answer:
[tex]n = 11[/tex]
Step-by-step explanation:
We are given two arithmetic sequences:
7, 12, 17, 22... and 27, 30, 33, 36...
And we want to determine n such that the nth term of each sequence is equivalent.
We can write a direct formula for each sequence. Recall that the direct formula for an arithmetic sequence is given by:
[tex]\displaystyle x_n = a + d(n-1)[/tex]
Where a is the initial term and d is the common difference.
The first sequence has an initial term of 7 and a common difference of 5. Hence:
[tex]\displaystyle x_n = 7 + 5(n-1)[/tex]
The second sequence has an initial term of 27 and a common difference of 3. Hence:
[tex]x_n = 27 + 3(n-1)[/tex]
Set the two equations equal to each other:
[tex]7 + 5 (n-1) = 27 + 3(n-1)[/tex]
Solve for n. Distribute:
[tex]7 + (5n - 5) = 27 + (3n - 3)[/tex]
Combine like terms:
[tex]5n + 2 = 3n + 24[/tex]
Isolate:
[tex]2n = 22[/tex]
Divide. Hence:
[tex]n = 11[/tex]
In conclusion, the 11th term of the first A.P. is equivalent to the 11th term of the second.
