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

Hint: 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.