Question

The value of 1+3+5+7+9+……+25 is:
A.196
B.625
C.225
D.169

Answer 1

Hint: In this question, we need to evaluate the value of the given expression 1+3+5+7+9+……+25. For this, we will first check about the nature of the progression and then, use the relation for the summation of the series of the progression.

Complete step-by-step answer:
The given series is 1+3+5+7+9+……+25. Here we can see that the difference is 2 in between every consecutive terms and so, we can say that the series is Arithmetic Progression series.
Now, using the summation formula for the arithmetic series which is given as:
 $ S = \dfrac{n}{2}\left[ {a + l} \right] $ where, ‘n’ is the total number of terms in the series, ‘a’ is the first term of the series and ‘l’ is the last term of the series.
For using this formula, we need to first calculate the total number of terms present in the given series by using the formula $ l = a + (n - 1)d $ where, ‘l’ is the last term of the series, ‘n’ is the total number of terms, ‘a’ is the first term of the series and ‘d’ is the common difference of the series.
Here, the first term 1, the last term is 25 and the common difference 2.
Substituting the values in the formula $ l = a + (n - 1)d $ to determine the total number of terms present in the given series.
 $
  l = a + (n - 1)d \\
   \Rightarrow 25 = 1 + (n - 1)2 \\
   \Rightarrow 2(n - 1) = 25 - 1 \\
   \Rightarrow n - 1 = \dfrac{{24}}{2} \\
   \Rightarrow n = 12 + 1 \\
   \Rightarrow n = 13 \;
  $
Now, substituting all the known values in the formula $ S = \dfrac{n}{2}\left[ {a + l} \right] $ to determine the summation of the given series.
 $
  S = \dfrac{n}{2}\left[ {a + l} \right] \\
   \Rightarrow S = \dfrac{{13}}{2}\left[ {1 + 25} \right] \\
   \Rightarrow S = \dfrac{{13 \times 26}}{2} \\
   \Rightarrow S = 13 \times 13 \\
   \Rightarrow S = 169 \;
  $
Hence, the value of the given series is 169.
So, the correct answer is “Option D”.

Note: Here in the question, according to the given series progression, we can see that the difference between any two consecutive terms is the same i.e., 2 and so, we have used the concept of the arithmetic progression only. There are other known progression series as well like Geometric progression (having common ratio) and Harmonic Progression.