Question

Find the first term and common difference of the A.P.
A.\[5,2, - 1, - 4, \ldots .\]
B. $ \dfrac{1}{2},\dfrac{5}{6},\dfrac{7}{6},\dfrac{3}{6},....,\dfrac{{17}}{6} $

Answer 1

Hint: The term from which an arithmetic progression begins is called the first term of the arithmetic progression and the difference between any two consecutive terms is called the common difference of that arithmetic progression. We are given two arithmetic progressions in the question, their first term can be found out easily and the common difference can be found out using the definition written above.

Complete step-by-step answer:
A.The arithmetic progression given is \[5,2, - 1, - 4, \ldots .\]
On observing this A.P. we see that the beginning term is 5, so the first term of this A.P. is 5
The common difference of the given A.P. is
 $
  d = 2 - 5 = - 4 + ( - 1) \\
   \Rightarrow d = - 3 \;
  $
Hence, the first term of the given A.P. is $ 5 $ and the common difference is $ -3 $ .

B.The arithmetic progression given is $ \dfrac{1}{2},\dfrac{5}{6},\dfrac{7}{6},\dfrac{3}{6},....,\dfrac{{17}}{6} $ .
The A.P. given above begins with $ \dfrac{1}{2} $ , so the first term of this A.P. is $ \dfrac{1}{2} $
The common difference of the given A.P. is
 $
  d = \dfrac{5}{6} - \dfrac{1}{2} = \dfrac{7}{6} - \dfrac{5}{6} \\
   \Rightarrow d = \dfrac{1}{3} \\
  $
Hence, the first term of the given A.P. is $ \dfrac{1}{2} $ and the common difference is $ \dfrac{1}{3} $ .

Note: A progression or sequence of numbers such that the difference between any two consecutive numbers is constant is called an Arithmetic progression. An A.P. consists of many terms, we can extend an A.P. up to infinite number of terms by finding its common difference and adding it to the previous term and so on, so it becomes important to find out the common difference of an A.P. because it is the most important thing in determining the A.P.