Answer
Verified
430.5k+ views
Hint: An arithmetic series is the sum of a sequence $\left\{a_{k}\right\}, k=1,2, \ldots,$ in which each term is computed from the previous one by adding (or subtracting) a constant d. Therefore, for $k>1$, $a_{k}=a_{k-1}+d=a_{k-2}+2 d=\ldots=a_{1}+d(k-1)$
The sum of the sequence of the first n terms is then given by:
${{S}_{n}}\text{ }\equiv \sum\limits_{k=1}^{n}{{{a}_{k}}}=\sum\limits_{k=1}^{n}{\left[ {{a}_{1}}+(k-1)d \right]}$
$\mathrm{Sn}=\mathrm{n}(\mathrm{a} 1+\mathrm{an}) 2$
$=n{{a}_{1}}+d\sum\limits_{k=1}^{n-1}{k}$
Complete step-by-step answer:
An arithmetic sequence is a sequence where the difference d between successive terms is constant. An arithmetic series is the sum of the terms of an arithmetic sequence. The nth partial sum of an arithmetic sequence can be calculated using the first and last terms as follows:
$\mathrm{Sn}=\mathrm{n/2}(\mathrm{a_1} +\mathrm{a_n}) $
An arithmetic sequence is a sequence with the difference between two consecutive terms constant. The difference is called the common difference. A
geometric sequence is a sequence with the ratio between two consecutive terms
constant.
Given series is $5+11+17+\ldots+95$.
Thus, $a=5, d=11-5=6, l=95$
$\mathrm{n}=\dfrac{\mathrm{l}-\mathrm{a}}{\mathrm{d}}+1$
$=\dfrac{95-5}{6}+1$
$=\dfrac{90}{6}+1=15+1=16$
Therefore, $\mathrm{S}_{\mathrm{n}}=\dfrac{\mathrm{n}}{2}(\mathrm{a}+\mathrm{l})$
$=\dfrac{16}{2}(5+95)$
$=8(100)$
$=800$
Note: Using the sum identity
$\sum_{k=1}^{n} k=\dfrac{1}{2} n(n+1)$ then gives $S_{n}=n a_{1}+\dfrac{1}{2} d n(n-1)=\dfrac{1}{2} n\left[2 a_{1}+d(n-1)\right]$
Note, however, that $a_{1}+a_{n}=a_{1}+\left[a_{1}+d(n-1)\right]=2 a_{1}+d(n-1)$
So $S_{n}=\dfrac{1}{2} n\left(a_{1}+a_{n}\right)$ or n times the arithmetic mean of the first and last terms.
The sum of the sequence of the first n terms is then given by:
${{S}_{n}}\text{ }\equiv \sum\limits_{k=1}^{n}{{{a}_{k}}}=\sum\limits_{k=1}^{n}{\left[ {{a}_{1}}+(k-1)d \right]}$
$\mathrm{Sn}=\mathrm{n}(\mathrm{a} 1+\mathrm{an}) 2$
$=n{{a}_{1}}+d\sum\limits_{k=1}^{n-1}{k}$
Complete step-by-step answer:
An arithmetic sequence is a sequence where the difference d between successive terms is constant. An arithmetic series is the sum of the terms of an arithmetic sequence. The nth partial sum of an arithmetic sequence can be calculated using the first and last terms as follows:
$\mathrm{Sn}=\mathrm{n/2}(\mathrm{a_1} +\mathrm{a_n}) $
An arithmetic sequence is a sequence with the difference between two consecutive terms constant. The difference is called the common difference. A
geometric sequence is a sequence with the ratio between two consecutive terms
constant.
Given series is $5+11+17+\ldots+95$.
Thus, $a=5, d=11-5=6, l=95$
$\mathrm{n}=\dfrac{\mathrm{l}-\mathrm{a}}{\mathrm{d}}+1$
$=\dfrac{95-5}{6}+1$
$=\dfrac{90}{6}+1=15+1=16$
Therefore, $\mathrm{S}_{\mathrm{n}}=\dfrac{\mathrm{n}}{2}(\mathrm{a}+\mathrm{l})$
$=\dfrac{16}{2}(5+95)$
$=8(100)$
$=800$
Note: Using the sum identity
$\sum_{k=1}^{n} k=\dfrac{1}{2} n(n+1)$ then gives $S_{n}=n a_{1}+\dfrac{1}{2} d n(n-1)=\dfrac{1}{2} n\left[2 a_{1}+d(n-1)\right]$
Note, however, that $a_{1}+a_{n}=a_{1}+\left[a_{1}+d(n-1)\right]=2 a_{1}+d(n-1)$
So $S_{n}=\dfrac{1}{2} n\left(a_{1}+a_{n}\right)$ or n times the arithmetic mean of the first and last terms.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
Difference Between Plant Cell and Animal Cell
Give 10 examples for herbs , shrubs , climbers , creepers
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE