Login/Sign Up
Home » Find the term of an A.P. sum of whose first n terms is 2n + 3n2. [Hint: ]
Find the r^{th} term of an A.P. sum of whose first n terms is 2n + 3n2. [Hint: a_{n} = S_{n} - S_{n}-1]
Guides
Find the r^{th} term of an A.P. sum of whose first n terms is 2n + 3n2. [Hint: a_{n} = S_{n} - S_{n}-1]

Solution:

The sum of the first n terms be {{S}_{n}}~=\text{ }2n\text{ }+\text{ }3{{n}^{2}}

Now we need to find the rth term i.e. ar

Utilizing the given hint nth term as an = Sn – Sn-1

\Rightarrow ~{{a}_{r}}~=\text{ }{{S}_{r}}~\text{-}{{S}_{r-1}}

Using {{S}_{n}}~=\text{ }2n\text{ }+\text{ }3{{n}^{2}}

\Rightarrow ~{{a}_{r}}~=\text{ }2r\text{ }+\text{ }3{{r}^{2}}~\text{-}(2\left( r\text{ }\text{-}1 \right)\text{ }+\text{ }3{{\left( r\text{ }\text{-}1 \right)}^{2}})

\Rightarrow ~{{a}_{r}}~=\text{ }2r\text{ }+\text{ }3{{r}^{2}}~\text{-}(2r\text{ }\text{-}2\text{ }+\text{ }3({{r}^{2}}~\text{-}2r\text{ }+\text{ }1))

\Rightarrow ~{{a}_{r}}~=\text{ }2r\text{ }+\text{ }3{{r}^{2}}~\text{-}(2r\text{ }\text{-}2\text{ }+\text{ }3{{r}^{2}}~\text{-}6r\text{ }+\text{ }3)

\Rightarrow ~{{a}_{r}}~=\text{ }6r\text{ }\text{-}1

Thus the rth term is 6r – 1

i

More Material