Answer
437.4k+ views
Hint: We are given the sum of m terms denoted by \[2{{m}^{2}}+3m,\] so we will put m = 1 and will get the sum of the first term which is nothing but the first term only. Then, we put m = 2 and get the sum of the first two terms of the AP. Finally, we will subtract the first term from the sum of the two terms to get the second term.
Complete step-by-step answer:
We are given that the sum of the first m term of the AP is given as \[2{{m}^{2}}+3m.\] We are asked to find the second term to do so we need the common difference and the first term.
Now, \[2{{m}^{2}}+3m\] denotes the sum of the first m terms. When we put m = 1, it means we get the sum of the first term only. So,
\[{{S}_{1}}=2\times {{1}^{2}}+3\times 1\]
\[\Rightarrow {{S}_{1}}=2+3\]
\[\Rightarrow {{S}_{1}}=5\]
So, the sum of the first term is 5. This means we have got \[{{a}_{1}}=5.\left[ {{a}_{1}}=\text{first term} \right]\]
Now, when we put m = 2 in \[2{{m}^{2}}+3m,\] we will get the sum of the first two terms of the AP. Therefore, we get,
\[{{S}_{2}}=2{{m}^{2}}+3m\]
\[\Rightarrow {{S}_{2}}=2\times {{2}^{2}}+3\times 2\]
Solving further, we get,
\[\Rightarrow {{S}_{2}}=2\times 4+6\]
\[\Rightarrow {{S}_{2}}=14\]
So, we get the sum of the first two terms as 14.
We will take \[{{a}_{1}}\] as the first term and \[{{a}_{2}}\] as the second term. Then we have,
\[{{a}_{1}}+{{a}_{2}}={{S}_{2}}\]
As \[{{S}_{2}}=14,{{a}_{1}}=5,\] we get,
\[\Rightarrow {{a}_{2}}=14-5\]
\[\Rightarrow {{a}_{2}}=9\]
Hence, we get that the second term of the AP is 9.
So, the required answer is 9.
Note: We can also find the second term by first finding ‘a’ (first term) and ‘d’ (common difference).
When we put m = 1, we get,
\[{{S}_{1}}=2{{\times }^{2}}+3\times 1\]
\[\Rightarrow {{S}_{1}}=5\]
So, our first term is 5. Therefore, a is 5.
Then in an AP, the second term is denoted by d. When we put m = 2, we get,
\[{{S}_{2}}=2\times {{2}^{2}}+3\times 2\]
\[\Rightarrow {{S}_{2}}=2\times 4+6\]
\[\Rightarrow {{S}_{2}}=14\]
\[{{S}_{2}}=\text{Sum of first term + second term}\]
\[\Rightarrow a+\left( a+d \right)=14\]
\[\Rightarrow 5+5+d=14\]
\[\Rightarrow 10+d=14\]
\[\Rightarrow d=14-10\]
\[\Rightarrow d=4\]
Now, we have a = 5 and d = 4. So, we can easily find the second term,
\[{{a}_{2}}=a+d\]
\[\Rightarrow {{a}_{2}}=5+4\]
\[\Rightarrow {{a}_{2}}=9\]
So, the second term is 9.
Complete step-by-step answer:
We are given that the sum of the first m term of the AP is given as \[2{{m}^{2}}+3m.\] We are asked to find the second term to do so we need the common difference and the first term.
Now, \[2{{m}^{2}}+3m\] denotes the sum of the first m terms. When we put m = 1, it means we get the sum of the first term only. So,
\[{{S}_{1}}=2\times {{1}^{2}}+3\times 1\]
\[\Rightarrow {{S}_{1}}=2+3\]
\[\Rightarrow {{S}_{1}}=5\]
So, the sum of the first term is 5. This means we have got \[{{a}_{1}}=5.\left[ {{a}_{1}}=\text{first term} \right]\]
Now, when we put m = 2 in \[2{{m}^{2}}+3m,\] we will get the sum of the first two terms of the AP. Therefore, we get,
\[{{S}_{2}}=2{{m}^{2}}+3m\]
\[\Rightarrow {{S}_{2}}=2\times {{2}^{2}}+3\times 2\]
Solving further, we get,
\[\Rightarrow {{S}_{2}}=2\times 4+6\]
\[\Rightarrow {{S}_{2}}=14\]
So, we get the sum of the first two terms as 14.
We will take \[{{a}_{1}}\] as the first term and \[{{a}_{2}}\] as the second term. Then we have,
\[{{a}_{1}}+{{a}_{2}}={{S}_{2}}\]
As \[{{S}_{2}}=14,{{a}_{1}}=5,\] we get,
\[\Rightarrow {{a}_{2}}=14-5\]
\[\Rightarrow {{a}_{2}}=9\]
Hence, we get that the second term of the AP is 9.
So, the required answer is 9.
Note: We can also find the second term by first finding ‘a’ (first term) and ‘d’ (common difference).
When we put m = 1, we get,
\[{{S}_{1}}=2{{\times }^{2}}+3\times 1\]
\[\Rightarrow {{S}_{1}}=5\]
So, our first term is 5. Therefore, a is 5.
Then in an AP, the second term is denoted by d. When we put m = 2, we get,
\[{{S}_{2}}=2\times {{2}^{2}}+3\times 2\]
\[\Rightarrow {{S}_{2}}=2\times 4+6\]
\[\Rightarrow {{S}_{2}}=14\]
\[{{S}_{2}}=\text{Sum of first term + second term}\]
\[\Rightarrow a+\left( a+d \right)=14\]
\[\Rightarrow 5+5+d=14\]
\[\Rightarrow 10+d=14\]
\[\Rightarrow d=14-10\]
\[\Rightarrow d=4\]
Now, we have a = 5 and d = 4. So, we can easily find the second term,
\[{{a}_{2}}=a+d\]
\[\Rightarrow {{a}_{2}}=5+4\]
\[\Rightarrow {{a}_{2}}=9\]
So, the second term is 9.
Recently Updated Pages
Mark and label the given geoinformation on the outline class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
When people say No pun intended what does that mea class 8 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Name the states which share their boundary with Indias class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give an account of the Northern Plains of India class 9 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Advantages and disadvantages of science
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
10 examples of evaporation in daily life with explanations
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
How do you graph the function fx 4x class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)