Question

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

Answer 1

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 + 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}$$.