Question

How do you find the number of terms $ n $ given the sum $ {S_n} = 182 $ of the series $ 50 + 42 + 34 + 26 + 18 $ …?

Answer 1

Hint: First we will specify all the given terms and then evaluate the values of the required terms from the question. Then we evaluate the value of common difference and solve for the value of $ n $ . We will be using the following formula here: $ {S_n} = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right] $ .

Complete step-by-step answer:
Now in these kinds of questions, we start by first mentioning the given terms first. So here, we can write,
 $
  {S_n} = 182 \\
  \,\,a = 50 \;
  $
Now we solve for the value of the common difference that is $ d $ .
 $ d = {a_2} - {a_1} $
Here, the value of $ {a_1} $ is $ 50 $ and the value of $ {a_2} $ is $ 42 $ .
Hence, the value of common difference will be,
 $
  d = {a_2} - {a_1} \\
  d = 42 - 50 \\
  d = - 8 \;
  $
Hence, the value of $ d $ is $ - 8 $ .
Now, we mention all the evaluated terms.
 $
  {S_n} = 182 \\
  \,\,a = 50 \\
  \,d = - 8 \;
  $
So, now substitute all these terms in our equation of sum of all the terms.
 $
  {S_n} = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right] \\
  182 = \dfrac{n}{2}\left[ {2(50) + (n - 1)( - 8)} \right] \\
  182 = \dfrac{n}{2}\left[ {(100) - (n - 1)(8)} \right] \\
  $
Now cross multiply the terms and take all the like terms to one side.
\[
  364 = n\left[ {100 - (8n - 8)} \right] \\
  364 = 100n - n(8n - 8) \\
  364 = 100n - 8{n^2} + 8n \\
  54n - 4{n^2} = 182 \;
 \]
Now solve the quadratic equation $ 4{n^2} - 54n + 182 = 0 $ by using the quadratic formula.
The quadratic formula is given by $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ .
Now we compare the value of terms from the general form of quadratic equation with our equation.
The general quadratic equation is given by $ a{x^2} + bx + c = 0 $ .
After comparison we get the values as,
 $
  a = 4 \\
  b = - 54 \\
  c = 182 \;
  $
Now, we apply the quadratic formula, to evaluate the value of $ n $ .
\[
  n = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} \\
  n = \dfrac{{ - ( - 54) \pm \sqrt {{{( - 54)}^2} - 4(4)(182)} }}{{2(4)}} \\
  n = \dfrac{{54 \pm \sqrt {2916 - 2912} }}{8} \\
  n = \dfrac{{54 \pm \sqrt 4 }}{8} \\
  n = \dfrac{{54 \pm 2}}{8} \\
 \]
Hence, the value of $ n $ will be
   $
  n = \dfrac{{54 + 2}}{8} = 7 \\
  n = \dfrac{{54 - 2}}{8} = 6.5 \;
  $
But $ n $ can only be a whole number, hence the value of $ n $ will be $ 7 $ .
So, the correct answer is “$ 7 $”.

Note: While solving such types of questions, begin by mentioning all the given terms first in order to avoid any confusions. When you solve for the value of common difference, make sure you substitute the first and second term properly. While applying the quadratic formula, make sure you substitute values of terms along with their signs.