Answer
64.8k+ views
Hint: Use the formula of Arithmetic progression sequence for the nth 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. Calculate the value of a. We use the formula of the sum of n terms in Arithmetic progression that is \[{S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]\]. Calculate the sum of the AP, \[{S_n}\].
Complete step by step solution:
Given data:The ${22^{nd}}$ term that is given for an arithmetic progression is 149.
Common difference is \[d = 7\]
Now, we know about the Arithmetic progression sequence for the nth terms is given by the following expression:
\[{a_n} = a + \left( {n - 1} \right)d\]
Here, the first term of the arithmetic progression sequence is $a$.
Now, calculate the value of $a$. Substitute the value of d = 7,n = 22 and ${a_n} = 149$ in \[{a_n} = a + \left( {n - 1} \right)d\].
149 = a + (22- 1)7
149 = a + 147
a = 149 - 147
= 2
Now, we know about the formula of the sum of n terms in Arithmetic progression is given by the following expression:
\[{S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]\]
Simplify the above equation by substituting \[{a_n} = a + \left( {n - 1} \right)d\].
\[{S_n} = \dfrac{n}{2}\left[ {a + {a_n}} \right]\]
Now, calculate the value of ${S_n}$ by substituting $n = 23$, $a = 2$ and $a_n = 149$ in the expression for the sum of the Arithmetic progression \[{S_n} = \dfrac{n}{2}\left[ {a + {a_n}} \right]\].
${S_{22}} = \dfrac{{22}}{2}\left[ {2 + 149} \right]\\
= 11\left[ {151} \right]\\
= 1,661
$
Hence, the sum of the first 22 terms of an Arithmetic progression is \[{S_{22}} = 1,661\].
Note: The general equation of the Arithmetic progression is \[a,a + d,a + 2d,a + 3d,...\], where a is initial term of the AP and d is the common difference of successive numbers. Make sure use the formula of the sum of n terms in Arithmetic progression that is \[{S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]\] and use the Arithmetic progression sequence for the nth terms that is \[{a_n} = a + \left( {n - 1} \right)d\].
Complete step by step solution:
Given data:The ${22^{nd}}$ term that is given for an arithmetic progression is 149.
Common difference is \[d = 7\]
Now, we know about the Arithmetic progression sequence for the nth terms is given by the following expression:
\[{a_n} = a + \left( {n - 1} \right)d\]
Here, the first term of the arithmetic progression sequence is $a$.
Now, calculate the value of $a$. Substitute the value of d = 7,n = 22 and ${a_n} = 149$ in \[{a_n} = a + \left( {n - 1} \right)d\].
149 = a + (22- 1)7
149 = a + 147
a = 149 - 147
= 2
Now, we know about the formula of the sum of n terms in Arithmetic progression is given by the following expression:
\[{S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]\]
Simplify the above equation by substituting \[{a_n} = a + \left( {n - 1} \right)d\].
\[{S_n} = \dfrac{n}{2}\left[ {a + {a_n}} \right]\]
Now, calculate the value of ${S_n}$ by substituting $n = 23$, $a = 2$ and $a_n = 149$ in the expression for the sum of the Arithmetic progression \[{S_n} = \dfrac{n}{2}\left[ {a + {a_n}} \right]\].
${S_{22}} = \dfrac{{22}}{2}\left[ {2 + 149} \right]\\
= 11\left[ {151} \right]\\
= 1,661
$
Hence, the sum of the first 22 terms of an Arithmetic progression is \[{S_{22}} = 1,661\].
Note: The general equation of the Arithmetic progression is \[a,a + d,a + 2d,a + 3d,...\], where a is initial term of the AP and d is the common difference of successive numbers. Make sure use the formula of the sum of n terms in Arithmetic progression that is \[{S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]\] and use the Arithmetic progression sequence for the nth terms that is \[{a_n} = a + \left( {n - 1} \right)d\].
Recently Updated Pages
Write a composition in approximately 450 500 words class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What is the common property of the oxides CONO and class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The area of square inscribed in a circle of diameter class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Other Pages
Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
A boat takes 2 hours to go 8 km and come back to a class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Electric field due to uniformly charged sphere class 12 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
According to classical free electron theory A There class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
In the ground state an element has 13 electrons in class 11 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)