Question

The solution of the equation $(x + 1) + (x + 4) + (x + 7) + ...... + (x + 28) = 155$is
$
  1)\;{\text{1}} \\
  {\text{2) 2}} \\
  {\text{3) 3}} \\
  4){\text{ 4}} \\
 $

Answer 1

Hint: First of all we will identify the pattern of the given series and then we will find the sum of all the terms and will simplify the equation to get the value of “x” making it the subject and moving all the terms on the opposite side.

Complete step-by-step answer:
Take the given expression: $(x + 1) + (x + 4) + (x + 7) + ...... + (x + 28) = 155$
Here we will check for the pattern of the series, find the difference of the consecutive terms –
$
  d = (x + 4) - (x + 1) \\
  d = x + 4 - x - 1 \\
  d = 3 \;
 $(Like terms with the same value and the opposite sign cancels each other)
Similarly, let us find the difference between the third and the fourth term –
$
  d = (x + 7) - (x + 4) \\
  d = x + 7 - x - 4 \\
  d = 3 \;
 $
Since, the difference between any two terms remains the same, therefore the given series is the arithmetic sequence.
Here, first term $a = x + 1$
Difference, $d = 3$
Nth term, ${a_n} = x + 28$
Nth term can be given by –
${a_n} = a + (n - 1)d$
Place the values in the above equation –
$x + 28 = x + 1 + (n - 1)3$
Simplifying the above expression, like terms with the same value and the same sign on opposite sides cancels each other.
$28 = 1 + (n - 1)3$
Make like terms on one side –
$
  28 - 1 = (n - 1)3 \\
  27 = 3(n - 1) \;
 $
Term multiplicative on one side if moved to the opposite side then it goes to the denominator –
$
  9 = n - 1 \\
  n = 9 + 1 \\
  n = 10 \;
 $
Now, $S_n^{} = \dfrac{n}{2}(a + l)$
Place the values in the above expression –
$155 = \dfrac{{10}}{2}(x + 1 + x + 28)$
Simplify the above expression –
$155(2) = 10(2x + 29)$
$\dfrac{{155(2)}}{{10}} = (2x + 29)$
Remove common factors from the numerator and the denominator for the term on the left hand side of the equation –
$31 = 2x + 29$
Make the required term the subject, when you move any term from one side to the opposite side then the sign of the terms also changes.
$
  2x = 31 - 29 \\
  2x = 2 \\
  x = 1 \;
 $
From the given multiple choices, option first is the correct one.
So, the correct answer is “Option 1”.

Note: Be careful about the sign convention and while moving any terms from one side to the opposite side. Positive terms become negative and vice-versa. Be good in multiples and the division and finding the common factors and removing them when represented in the form of the fractions.