Question

The sum of n term of the series is
\[{\text{1}}{\text{.4 + 3}}{\text{.04 + 5}}{\text{.004 + 7}}{\text{.0004}}\]
A.\[{{\text{n}}^{\text{2}}}{\text{ + }}\dfrac{{\text{4}}}{{\text{9}}}{\text{(1 + }}\dfrac{{\text{1}}}{{{\text{1}}{{\text{0}}^{\text{n}}}}}{\text{)}}\]
B.\[{{\text{n}}^{\text{2}}}{\text{ + }}\dfrac{{\text{4}}}{{\text{9}}}{\text{(1 - }}\dfrac{{\text{1}}}{{{\text{1}}{{\text{0}}^{\text{n}}}}}{\text{)}}\]
C.\[{{\text{n}}^{\text{2}}}{\text{ + }}\dfrac{{\text{4}}}{{\text{9}}}{\text{(1 + }}\dfrac{{\text{1}}}{{{\text{1}}{{\text{0}}^{\text{n}}}}}{\text{)}}\]
D.None of these

Answer 1

Hint: The above given series can be rearranged and written as the sum of A.P and G.P as
\[{\text{1 + 3 + 5}}...{\text{and 0}}{\text{.4 + 0}}{\text{.04 + 0}}{\text{.004}}...\] And so now proceed with sum of terms of A.P and G.P as \[\dfrac{{\text{n}}}{{\text{2}}}{\text{(2a + (n - 1)d)}}\] and \[\dfrac{{{\text{a(}}{{\text{r}}^{\text{n}}}{\text{ - 1)}}}}{{{\text{r - 1}}}}\]

Complete step-by-step answer:
The given series is given as \[{\text{1}}{\text{.4 + 3}}{\text{.04 + 5}}{\text{.004 + 7}}{\text{.0004}}\]
As the above series can be written as
\[{\text{1 + 3 + 5 + }}...{\text{ + 0}}{\text{.4 + 0}}{\text{.04 + 0}}{\text{.004 + }}..\]
And thus we can see in the above equation that they are summation of both A.P and G.P as
In\[{\text{A}}{\text{.P,a = 1,d = 2}}\]
In \[{\text{G}}{\text{.P,a = 0}}{\text{.4,r = 0}}{\text{.1}}\]
Substituting all the values in the above general equation of A.P and G.P
So we obtain,
\[
   \Rightarrow \dfrac{n}{2}[2(1) + (n - 1)2] + \dfrac{{{\text{0}}{\text{.4((0}}{\text{.1}}{{\text{)}}^{\text{n}}}{\text{ - 1)}}}}{{{\text{(0}}{\text{.1) - 1}}}} \\
   \Rightarrow \dfrac{n}{2}(2n) + \dfrac{{{\text{0}}{\text{.4((1 - }}\dfrac{1}{{{{10}^n}}}{\text{)}}}}{{0.9}} \\
   \Rightarrow {n^2} + \dfrac{4}{9}({\text{1 - }}\dfrac{1}{{{{10}^n}}}) \\
 \]
Hence , option (a) is our required correct answer.

Note: An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant. For example, the sequence 1, 2, 3, 4, ... is an arithmetic progression with common difference 1
A geometric progression, also known as a geometric sequence, is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ratio 3.