Question

Find the next term of the AP\[\sqrt {8\,} ,\sqrt {12} \,\sqrt {32} \].....

Answer 1

 Hint: A series is said to be in an arithmetic sequence in which the difference between each consecutive term of the sequence is constant. In general, the AP series is represented by the formula\[{t_n} = a + \left( {{\text{n - 1}}} \right)d\] where $n$ the number of terms.
The behaviour of the series depends on the common difference, if it is positive then, the sequence is increasing towards the infinity, and if the difference is negative, then the series is decreasing to the negative infinity.

Complete step-by-step answer:
In general, an AP series is represented as \[a,a + d,a + 2d,a + 3d,.......\]
Here, the first term of the sequence is$\sqrt 8 $, which can also be written as:
\[
  a = \sqrt {8\,} \\
   = 2\sqrt 2 \\
 \]
Similarly, the second term of the sequence is$\sqrt {12} $, which can also be written as:
\[
  a + d = \sqrt {12} \\
   = 3\sqrt 2 \\
 \]
Now, to calculate the difference $(d)$ of the AP series, the difference between the first term and the second term needs to be calculated as:
\[
  d = (a + d) - a \\
   = 3\sqrt 2 - 2\sqrt 2 \\
   = \sqrt 2 \\
 \]

As the common difference $(d)$ is positive, the AP sequence in an increasing sequence and its fourth term will be greater than the third term of the sequence.

Now to find the 4th term of the sequence, i.e., n=4 the 4th term of the AP series, substitute$a = 2\sqrt 2 $, $d = \sqrt 2 $ and $n = 4$ in the formula ${t_n} = a + (n - 1)d$
\[
  {t_4} = a + (4 - 1)d \\
   = a + 3d \\
   = 2\sqrt 2 + 3\sqrt 2 \\
   = 5\sqrt 2 \\
   = \sqrt {50} \\
 \]
Hence, the next term of the AP series is\[\sqrt {50} \]. By finding the fourth term of the sequence, we can say the AP sequence is increasing as\[\sqrt {50} > \sqrt {32} \].

Note: Before finding any term of a sequence, we need to find its common difference \[d\] because it helps to know the nature of the series as it tells whether the missing term is greater or lesser than its previous terms. The common difference is positive for increasing sequence and negative for decreasing sequence.