Question

Find the sum of the first 51 terms of an AP whose second and third terms are 14 and 18.

Answer 1

Hint: TO solve this problem we will use the sum of n terms formula. The formula of Arithmetic progression sequence for the $n^{th}$ terms that is \[{a_n} = a + \left( {n - 1} \right)d\] where, a initial term of the AP and d is the common difference of successive numbers. After that substitute the value of n which is given.

Complete step-by-step answer:
Given data:
The second and third terms of the Arithmetic progression are 14 and 18.
Now, we know about the Arithmetic progression sequence for the $n^{th}$ term is \[{a_n} = a + \left( {n - 1} \right)d\].
Now, calculate the value of ${a_n}$ where $n = 2$. Substitute the value of $n = 2$ and \[{a_n} = 14\] in the expression \[{a_n} = a + \left( {n - 1} \right)d\].
\[14 = a + \left( {2 - 1} \right)d\\
14 = a + d\\
a = 14 - d\] -----(i)
Now, calculate the value of ${a_n}$ where $n = 3$. Substitute the value of $n = 3$ and \[{a_n} = 18\] in the expression \[{a_n} = a + \left( {n - 1} \right)d\].
\[18 = a + \left( {3 - 1} \right)d\\
18 = a + 2d\\
a = 18 - 2d\]-----(ii)
Now, subtract the equation (i) from equation (ii) and obtain the value of d:
\[14 - d = 18 - 2d\\
d = 4\]
Calculate the value of a by substituting the value of d in the equation (i).
\[a = 14 - 4\\
a = 10\]
Now, we know about the formula of the sum of n terms in Arithmetic progression that is given by the following expression \[{S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]\].
Now, calculate the value of ${S_n}$ where $n = 51,a = 10,{\rm{ and }}d = 4$. Substitute the values in the expression \[{S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]\].
\[{S_{51}} = \dfrac{{51}}{2}\left[ {2\left( {10} \right) + \left( {51 - 1} \right)4} \right]\\
 = 25.5\left[ {220} \right]\\
 = 5,610\]

Hence, the sum of the first 51 terms of an Arithmetic progression is \[{S_{51}} = 5,610\].

Note: Here we learn about the Arithmetic mean, if \[a,b,c\] is in arithmetic progression, then the arithmetic mean is expressed by \[b = \dfrac{{a + c}}{2}\] where b is called the arithmetic mean of a and c. the general formula of the arithmetic for n positive numbers \[{a_1},{a_2},{a_3},...,{a_n}\] is given by $\text{Arithmetic mean}$ = \[\dfrac{{{a_1} + {a_2} + {a_3} + ... + {a_n}}}{n}\].