Question

How do you find the sum of the first $ 15 $ terms of an arithmetic sequence if the first term is $ 51 $ and the $ 25th $ term is 99?

Answer 1

Hint: An Arithmetic Progression (AP) is the sequence of numbers in which the difference of two successive numbers is always constant.
The standard formula for Arithmetic Progression is: $ {a_n} = a + (n - 1)d $
Where $ {t_n} = $ nth term in the AP
 $ a = $ First term of AP
 $ d = $ Common difference in the series
 $ n = $ Number of terms in the AP
Here we will find the first and twenty-fifth value in the standard formula and then will simplify both the equations and then will find the sum of the first fifteen terms of Arithmetic Sequence by using the formula.

Complete step by step solution:
First term, $ a = 51 $ (Given) …. (A)
Now, twenty-fifth term of an arithmetic progression –
 $ {a_{25}} = a + (25 - 1)d $
Simplify the above equation and place the given value –
 $ \Rightarrow 99 = a + 24d $
Place value from the equation (A)
 $ \Rightarrow 99 = 51 + 24d $
Move the constant term from the left hand side of the equation on the right hand side of the equation. When you move any term from one side to another sign also changes. Positive terms become negative and vice-versa.
 $ \Rightarrow 99 - 51 = 24d $
Simplify the above expression-
 $ \Rightarrow 48 = 24d $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator.
 $ \Rightarrow d = \dfrac{{48}}{{24}} $
Simplify-
 $ \Rightarrow d = 2 $ ….. (B)
Now, the sum of first fifteen terms can be calculated by using the formula –
 $ {S_n} = \dfrac{n}{2}[2a + (n - 1)d] $
By using equations (A) and (B)
 $ {S_{15}} = \dfrac{{15}}{2}[2(51) + (15 - 1)2] $
Simplify the above expression-
 $ {S_{15}} = \dfrac{{15}}{2}[102 + 28] $
Simplify –
 $
  {S_{15}} = \dfrac{{15}}{2}[130] \\
  {S_{15}} = 15 \times 65 \\
  {S_{15}} = 975 \;
  $
This is the required solution.
So, the correct answer is “975”.

Note: Be careful about the signs of the terms while simplification. When the terms are moved from one side to another, Sign of the term is also changed. Positive terms become negative and vice-versa. When you add one positive and one negative, you have to subtract and give a sign of the bigger number.