Question

Find the sum:
\[\left( { - 5} \right) + \left( { - 8} \right) + \left( { - 11} \right) + ....... + \left( { - 230} \right)\]

Answer 1

Hint: To solve this question first we assume a variable equal to the sum of the given expression and find that this expression is in arithmetic progression. Then we find the number of terms by the formula of the nth term of arithmetic progression. Then use the sum of arithmetic progression to find the sum of the given expression.

Complete step-by-step solution:
Let \[x = \left( { - 5} \right) + \left( { - 8} \right) + \left( { - 11} \right) + ....... + \left( { - 230} \right)\]
The expression given is in arithmetic progression
First term of an arithmetic progression is \[a = - 5\]
Common difference of the arithmetic progression is \[{\text{second term}} - {\text{first term}}\]
\[d = \left( { - 8} \right) - \left( { - 5} \right) =-3 \]
Last term of the common difference is
\[l = - 230\]
From here first we find the number of terms.
Using the formula of the nth term of an arithmetic progression.
\[{a_n} = a + \left( {n - 1} \right) \times d\]
On putting all the values
\[ - 230 = - 5 + \left( {n - 1} \right) \times - 3\]
Of taking \[ - 5\] to another side
\[ - 230 + 5 = \left( {n - 1} \right) \times - 3\]
On takin \[ - 3\] on denominator
\[\dfrac{{ - 225}}{{ - 3}} = \left( {n - 1} \right)\]
If we divide two negative numbers then the answer is also a negative number.
\[\left( {n - 1} \right) = 75\]
On further solving we get the number of terms
\[n = 76\]
Now we have to find the sum of the given expression
Using the formula of sum of an arithmetic progression we get the value of sum of that terms
\[{s_n} = \dfrac{n}{2}\left( {a + {a_n}} \right)\]
Here \[{s_n}\] is the sum of the n terms of an arithmetic progression.
On putting all the values.
\[{s_n} = \dfrac{{76}}{2}\left( { - 5 + - 230} \right)\]
On further solving.
\[{s_n} = 38\left( { - 235} \right)\]
On multiplying both the numbers.
\[{s_n} = 8930\]
Final answer:
The sum of the given expression is
\[ \Rightarrow {s_n} = 8930\]

Note: Students commit mistakes in finding that the given expression is in A.P or not and if they don’t know all the formulas so that they are unable to find the number of terms. It might be possible that the middle term of the given expression is given, then we find a number of terms for that number and make that double to get the last term.