Question

Find the sums indicated below:
\[3 + 6 + 9 + ... + 300\]

Answer 1

Hint: Here in this question, we have a sequence and we have to check whether the sequence belongs to an arithmetic sequence or not. We check the sequence with the help of arithmetic sequence definition and if it is an arithmetic sequence we determine the common difference of the sequence. Then we will determine the summation.

Complete step-by-step answer:
In the sequence we have three kinds of sequence namely, arithmetic sequence, geometric sequence and harmonic sequence.
In arithmetic sequence we the common difference between the two terms, In geometric sequence we the common ratio between the two terms, In harmonic sequence it is a ratio of arithmetic sequence to geometric sequence.
The general arithmetic progression is of the form \[a,a + d,a + 2d,...\] where a is first term nth d is the common difference. The nth term of the arithmetic progression is defined as \[{a_n} = {a_0} + (n - 1)d\]
Consider the given series
\[3 + 6 + 9 + ... + 300\]
The first term is \[{a_0} = 3\], the common difference is \[d = 3\] and \[{a_n} = 300\]. By this we have to determine the value of n.
By using the formula \[{a_n} = {a_0} + (n - 1)d\]
On substituting the value we have
\[ \Rightarrow 300 = 3 + (n - 1)3\]
on simplifying we get
\[ \Rightarrow 300 = 3 + 3n - 3\]
\[ \Rightarrow 300 = 3n\]
On dividing by 3 we get
\[ \Rightarrow n = 100\]
We have determined the value of n. Now we have to determine the summation
The summation formula for the arithmetic series is given as
\[{S_n} = \dfrac{n}{2}[2{a_0} + (n - 1)d]\]
On substituting the values we have
\[ \Rightarrow {S_{100}} = \dfrac{{100}}{2}[2(3) + (100 - 1)3]\]
On simplifying we have
\[ \Rightarrow {S_{100}} = 50[6 + 297]\]
On adding 6 and 297 we get
\[ \Rightarrow {S_{100}} = 50[303]\]
On multiplying 50 and 303 we have
\[ \Rightarrow {S_{100}} = 15150\]
Hence determined the value.
So, the correct answer is “\[ {S_{100}} = 15150\]”.

Note: By considering the formula of arithmetic sequence we verify the common difference which we obtained. We have to check the common difference for all the terms. Suppose if we check for the first two terms not for other terms then we may go wrong. So definition of arithmetic sequence is important to solve these kinds of problems.