Question

If the numbers $ a,9,b,25 $ form an AP, find a and b.

Answer 1

Hint: In order to determine the value of a and b , use the fact of the arithmetic sequence that the difference between any two consecutive terms in any AP is the same. So we can write $ 9 - a = b - 9 = 25 - b $ .Now use $ b - 9 = 25 - b $ to find the value of b and put this value of b in $ 9 - a = b - 9 $ to get the value of a.

Complete step-by-step answer:
Clearly, the sequence we are given is in Arithmetic Progression (A.P.) = $ a,9,b,25 $
Since , as we know, the difference between any two consecutive terms in any AP is the same.
So we can write
 $ 9 - a = b - 9 = 25 - b $ -----(1)
 $
  b - 9 = 25 - b \\
  b + b = 25 + 9 \\
  2b = 34 \;
  $
Dividing both sides of the equation with coefficient of b
 $
  \dfrac{{2b}}{2} = \dfrac{{34}}{2} \\
  b = 17 \;
  $
Putting $ b = 17 $ in equation (1)
 $
  9 - a = b - 9 \\
  9 + 9 = a + b \\
  a + b = 18 \\
  a + 17 = 18 \\
  a = 18 - 17 \\
  a = 1 \;
  $
 $ \therefore a = 1,b = 17 $
Therefore, the value of a and b is equal to 1 and 17 respectively.
So, the correct answer is “a=1 and b=17”.

Note: 1.Sequence: A sequence is a function whose domain is the set of N of natural numbers.
2.Real Sequence: A sequence whose range is a subset of R is called a real sequence.
In other words, a real sequence is a function having domain N and range equal to a subset of the set R of real numbers.
3.Arithmetic Progression (A.P): A sequence is called an arithmetic progression if the difference of a term and the previous term is always the same.
i.e. $ {a_{n + 1}} - {a_n} = $ constant $ ( = d) $ for all $ n \in N $ .
The constant difference is generally denoted by d which is called as the common difference.
In order to determine whether a sequence is an A.P. or not when its nth term is given, we may use the following algorithm .
Algorithm:
Step 1: Obtain $ {a_n} $ .
Step 2: Replace $ n $ by $ n + 1 $ in $ {a_n} $ to get $ {a_n} + 1 $
Step 3: Calculate $ {a_{n + 1}} - {a_n} $ .
Step 4: If $ {a_{n + 1}} - {a_n} $ is independent of n , the given sequence is an A.P.Otherwise is not an A.P. .
1.Don’t forgot to cross-check your answer.
2.The difference between any two consecutive terms in an A.P. is always the same and if it is not the same, then the given series is not an A.P.
3. Since the sequence does not contain the last term, it is an infinite series .