Two aps have the same common difference. the first term of one of these is –1 and that of the other is – 8. the difference between their 4th terms is

General
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two aps have the same common difference. the first term of one of these is –1 and that of the other is – 8. the difference between their 4th terms is

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LectureNotes said two APs have the same common difference. The first term of one of these is -1, and that of the other is -8. What is the difference between their 4th terms?

Answer:
Since both arithmetic progressions (APs) have the same common difference, we know that the 4th term of an AP where the first term is -1 can be calculated as:
[ a_{n} = a_{1} + (n-1)d ]
where:

  • ( a_{n} ) is the ( n^{th} ) term,
  • ( a_{1} ) is the first term,
  • ( n ) is the term number, and
  • ( d ) is the common difference.

For the first AP with ( a_{1} = -1 ), the 4th term is:
[ a_{4} = -1 + (4-1)d = -1 + 3d ]

For the second AP with ( a_{1} = -8 ), the 4th term is:
[ a_{4} = -8 + 3d ]

The difference between their 4th terms can be calculated by subtracting the first AP’s 4th term from the second AP’s 4th term:
[ (-8 + 3d) - (-1 + 3d) = -8 + 3d + 1 - 3d = -7 ]

Therefore, the difference between the 4th terms of these two APs is -7.