Question

Solve the equation: -4 + (-1) + 2 +……+ x = 437

Answer 1

Hint: In the type of question mentioned above we need to find the value of x in this we need to observe the given equation and we will be able to see that the given equation is an increasing AP, so by using the formula for the sum of AP we will find the number of terms in an AP and with that we will find the last term of the AP.

Complete step by step solution:
In the above mentioned question we can clearly see that the above equation is an increasing AP with the first term being -4 now to take out the difference we are going to subtract $1^{st}$ term from second term this will give us the difference now to check whether this equation is an AP or not we are going to again subtract $2^{nd}$ term form the $3^{rd}$ term and if the difference is same then we can conclude that the given equation is an AP (arithmetic progression). Now to check this we will first subtract $1^{st}$ term from $2^{nd}$ term and we will get:
D= -1-(-4)
D=3
Now to check whether this is an AP we are going to subtract $2^{nd}$ term from third and we will get:
D=2-(-1)
D=3
As we can see that the difference is same this is an AP. Now to calculate the last term of an AP we are going to use the formula of sum of AP which is based on Last term i.e.
\[sum=\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)\] Where a is the first term, n is the number of terms and d is the difference between the terms.
We know the sum to be 437, first term to be -4, and difference to be 3 now we will substitute this in the equation and we will get:
\[\begin{align}
  & \Rightarrow 437=\dfrac{n}{2}\left( 2\left( -4 \right)+\left( n-1 \right)3 \right) \\
 & \Rightarrow 437=\dfrac{n}{2}\left( -8+3n-3 \right) \\
 & \Rightarrow 874=n\left( -11+3n \right) \\
 & \Rightarrow 3{{n}^{2}}-11n-874=0 \\
\end{align}\]
On solving this quadratic equation we will get the value of n as 19.
So $19^{th}$ term of the equation is x as mentioned in the question.
 Now we will use the value of n to find the value of x. For this we will use the formula.
\[\begin{align}
  & x=a+\left( 19-1 \right)d \\
 & \Rightarrow x=-4+18\times 3 \\
 & \Rightarrow x=50 \\
\end{align}\]
So the value of x is 50.

Note: In the above type of question the given equations are generally a series of AP, GP and HP so always remember to check whether the equation is in any of the format as we know the general formulas which can help us solve the question faster and easier.