Question

The \[{50^{th}}\] term of the sequence \[3 + 12 + 25 + 42 + ....\] is
A.5145
B.5148
C.5142
D.5195

Answer 1

Hint: Given is a simple series given such that we call it quadratic number series. This is because the difference between two consecutive terms is when taken at second level will give us the same difference. And in these types of cases the \[{n^{th}}\] is given by a formula \[{T_n} = a{n^2} + bn + c\] . We also need to find the value of a, b and c along with that of \[{50^{th}}\] . We will use the values of the first three terms from the sequence given to get the values of a, b and c.

Complete step by step solution:
Given that the sequence is \[3 + 12 + 25 + 42 + ....\]
\[{T_n} = a{n^2} + bn + c\] is the general term.
Now we can see that the first term is 3. So we can put n equals to 1 and can write \[3 = a + b + c\] ……I
Next second term is 12. So we can put n equals to 2 and we can write \[12 = 4a + 2b + c\] ……II
And the third term is 25. So we can put n equals to 3 and we can write \[25 = 9a + 3b + c\] ……III
Now we have three variables and three equations. So we will solve them to get the values of a, b and c.
To find the values of the variables:
First we will find the value of c from equation I,
 \[c = 3 - a - b\]
Now putting this value on equation II and III
 \[12 = 4a + 2b + 3 - a - b\]
This is equation II which on solving will be
 \[9 = 3a + b\] …..IV
And equation III becomes \[25 = 9a + 3b + 3 - a - b\] which on solving will be
 \[22 = 8a + 2b\]
On dividing both sides by 2 we get,
 \[11 = 4a + b\] ….V
Now using equation IV and V we will subtract equation IV from V we get,
 \[11 - 9 = 4a + b - 3a - b\]
On solving we get,
 \[2 = a\]
That is the value of a.
Now putting this value in any of the equation IV or V we get,
 \[11 = 4 \times 2 + b\]
 \[11 - 8 = b\]
On solving we get, \[b = 3\]
This is the value of another variable.
Now putting these two values in equation I we get the value of c.
 \[3 = 2 + 3 + c\]
On solving we get
 \[c = - 2\]
This is the completion of one part of the solution. Now let’s find the value of \[{50^{th}}\] term.
 \[{T_{50}} = a{50^2} + b50 + c\]
Putting the values of a,b and c
 \[{T_{50}} = 2 \times 2500 + 3 \times 50 - 2\]
On solving we get,
 \[{T_{50}} = 5000 + 150 - 2\]
 \[{T_{50}} = 5148\]
This is the final answer. So option B is the correct answer.
So, the correct answer is “Option B”.

Note: Here note that problem is lengthy but it is too easy to solve. If in the problem we are given with the values of a, b and c then it is too easy. Simply need to use the formula and solve.
How one can determine whether the given series is a quadratic series or not then we will first find the difference in the given numbers of the series. Then we will find the difference in the consecutive numbers of the difference so obtained so if this second difference is the same then it is a quadratic series.