Question

The sum of series $1.2 + 2.3 + 3.4 + ......$10 terms is
a.440
b.286
c.524
d.$\infty$

Answer 1

This is a general problem of finding the sum of series where we are finding the sum of 10 terms of a given series. We will start with considering the general term, ${T_{n}} = n\left( {n + 1} \right)$, then proceed generally to find the formula of the sum of the terms in a general form, and then using that we calculate sum of 10 terms.

Complete step-by-step answer:
We have to find,
$1.2 + 2.3 + 3.4 + ......$10 terms
The general terms of the above series can be written as,
${T_{n}} = n\left( {n + 1} \right)$
$\therefore {S_n} =$ ​Sum of n terms of the series.
Now, we get
${S_n} = \sum {T_n}$
$= \sum n(n + 1)$
On Multiplying, we get
$= \sum ({n^2} + n)$
On opening bracket we get,
$= \sum {n^2} + \sum n$
As, sum of squares of 1st n terms, $= \dfrac{{n(n + 1)(2n + 1)}}{6}$ and sum of first n terms, $= \dfrac{{n(n + 1)}}{2}$, we get
$= \dfrac{{n(n + 1)(2n + 1)}}{6} + \dfrac{{n(n + 1)}}{2}$
On taking $\dfrac{{n(n + 1)}}{2}$ common we get,
$= \dfrac{{n(n + 1)}}{2}[\dfrac{{2n + 1}}{3} + 1]$
Now, if we simplify, we get
$= \dfrac{{n(n + 1)}}{2}[\dfrac{{2n + 1 + 3}}{3}]$
$= \dfrac{{n(n + 1)}}{2}[\dfrac{{2n + 4}}{3}]$
On taking 2 common from second term,
$= \dfrac{{n(n + 1)}}{2}.\dfrac{{2(n + 2)}}{3}$
On further simplification we get,
$= \dfrac{{n(n + 1)(n + 2)}}{3}$
Hence, ${S_n} = \dfrac{{n(n + 1)(n + 2)}}{3}$
As here, we are finding terms of 10 terms, we get, $n = 10$
We have, ${S_n} = \dfrac{{10(10 + 1)(10 + 2)}}{3}$

On simplification we get,
$= \dfrac{{10.11.12}}{3}$
On division we get,
$= 10.11.4$
On multiplication we get,
$= 440$
Hence, The sum of series $1.2 + 2.3 + 3.4 + ......$ up to 10 terms is 440
Hence, option (a) is correct.

Note: Here some of the things should be taken care of, that a sum of series of numbers will give us a result only when the series is convergent. If the series is not convergent we will not get a finite sum.
A series is convergent if the sequence of its partial sums tends to a limit; that means that the partial sums become closer and closer to a given number when the number of their terms increases.