Question

Solve the equation for x:
1 + 4 + 7 + 10 + ………. + x = 287

Answer 1

Hint: Find the common difference of the first term of the series. Substitute these values in the formula of sum of n terms and find the number of terms in the series. Thus find the value of x by substituting values in the formula of sum of n terms.
Complete step by step answer:
The given expression is an arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant. Now, considering the series,
1 + 4 + 7 + 10 + ………. + x = 287
The common difference, d = \[{{1}^{st}}\] term – \[{{2}^{nd}}\] term = 4 – 1 = 3
d = \[{{3}^{rd}}\] term - \[{{2}^{nd}}\] term = 7 – 4 = 3
Thus the common difference, d = 3 and first term = 1.
We have been given the sum of n terms as 287.
We know the formula to find sum of n terms in AP.
\[{{S}_{n}}=\dfrac{n}{2}\left[ 2a+\left( n-1 \right)d \right]\]
Let us find the value of n.
\[\begin{align}
  & 287=\dfrac{n}{2}\left[ 2\times 1+\left( n-1 \right)\times 3 \right] \\
 & 287\times 2=n\left[ 2+3n-3 \right] \\
 & 574=n\left[ 3n-1 \right] \\
 & \Rightarrow 574=3{{n}^{2}}-n \\
 & \therefore 3{{n}^{2}}-n-574=0 \\
\end{align}\]
The above expression is similar to the quadratic equation whose general equation is \[a{{x}^{2}}+bx+c=0\]. Let us compare both the equations. We get,
a = 3, b = -1, c = -574
Let us apply all these values in the formula to get n.
\[\begin{align}
  & n=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}=\dfrac{-\left( -1 \right)\pm \sqrt{{{\left( -1 \right)}^{2}}-4\times 3\times \left( -574 \right)}}{2\times 3} \\
 & n=\dfrac{1\pm \sqrt{1+6888}}{6}=\dfrac{1\pm \sqrt{6889}}{6}=\dfrac{1\pm 83}{6} \\
\end{align}\]
Thus \[n=\dfrac{1+83}{6}=\dfrac{84}{6}=14\] and \[n=\dfrac{1-83}{6}=\dfrac{-82}{6}=-13.67\].
Thus let us take n = 14.
We need to find x, which is the last term of the series. Then substitute the values in the equation of sum of n terms in AP.
\[{{S}_{n}}=\dfrac{n}{2}\left( a+l \right)\]
\[{{S}_{n}}\] = 287, n = 14, a = 1, l = x.
\[\begin{align}
  & 287=\dfrac{14}{2}\left( 1+x \right) \\
 & \dfrac{287\times 2}{14}=1+x \\
 & \therefore 1+x=41\Rightarrow x=41-1=40 \\
\end{align}\]
Thus the last term of the expression is x = 40.

Note: We have neglected n = -13.67, as it is a negative number and the number of terms in a series can never be negative. You should be able to identify the series whether it is AP, GP or HP and then apply the necessary formula.