Question

Calculate the value of $N$ in the given series and then find the value of $x$ using the given equation. $99\,\,163\,\,N\,\,248\,\,273\,\,289$. If $\sqrt {2N + 17} = x$, Then find the value of $x$.

Answer 1

According to the question, the series is $99\,\,163\,\,N\,\,248\,\,273\,\,289$. Here, we need to find out the value of $N$. The difference between the previous number and next number in the given series is the square values of $8,\,7,\,6,\,5\,and\,4$ respectively.

Complete step-by-step solution:
The difference between the respective numbers are the square values of $8,\,7,\,6,\,5\,and\,4$.
So, from the left-hand side of the series, the given number is $99$. So, if we add the square of $8$to $99$ then, we get:
$\Rightarrow 99 + {8^2} = 163$
Similarly, if we add the square of $5$ with $248$ then, we get:
$\Rightarrow 248 + {5^2} = 273$.
So, now if we add the square of $7$ with $163$, it will be $N$. This is now written as:
$\Rightarrow 163 + {7^2} = N$
$\Rightarrow N = 163 + 49 = 212$
Therefore, the value of $N$ in the sequence is $212$.
Now, we have to calculate the value of $x$ from the given equation, $\sqrt {2N + 17} = x$. Here, we will put the value of $N$.
Given: $\sqrt {2N + 17} = x$
$\Rightarrow \sqrt {(2 \times 212 + 17)} = x$
$\Rightarrow \sqrt {(424 + 17)} = x$
$\Rightarrow x = \sqrt {441}$
$\Rightarrow x = 21$
Therefore, the value of $x = 21$.

Note: In Mathematics, we say that a sequence is an ordered list of objects. This sequence has members, just like a set (also called elements or terms). Unlike a set, order matters, and a term may appear multiple times in the series at different points.