Question

Number of identical terms in the sequence $2,5,8,11,$ ___ upto 100 terms and $3,5,7,9,11$____ upto $100$ terms are
(A) $17$
(B) $33$
(C) $50$
(D) $147$

Answer 1

Hint: An arithmetic progression (AP) or arithmetic sequence is a set of numbers with a constant difference between consecutive terms. A finite arithmetic progression, also known as an arithmetic progression, is a finite component of an arithmetic progression. An arithmetic series is the sum of a finite arithmetic progression.

Complete step by step solution:
Let us consider the first series given here.
$2,5,8,11,$……….
Here, the first term ${a_1} = 2$ and the common difference ${\text{d}}{\text{ = }}{\text{5}}{\text{ - }}{\text{2}}{\text{ = }}{\text{3}}$
Number of terms, ${\text{n}}{\text{ = }}{\text{100}}$
Let the last term of the series be, ${{\text{a}}_{\text{n}}}$
The formula for the arithmetic progression is given by,
\[{{\text{a}}_{\text{n}}}{\text{ = }}{\text{a}}{\text{ + }}\left( {{\text{n - 1}}} \right){\text{d}}\]
Substituting the values in the above equation we get,
\[
  {a_n} = 2 + \left( {100 - 1} \right) \times 3 \\
   = 2 + 297 \\
  {a_n} = 299 \\
 \]
Hence, the series $ = 2,5,8,11,....................,197,200,203,.............,299$
Now consider the second series given.
$3,5,7,9,11$,………….
Here, the first term ${a_1} = 3$ and the common difference ${\text{d}}{\text{ = }}5 - 3 = 2$
Number of terms, ${\text{n}}{\text{ = }}{\text{100}}$
Let the last term of the series be, ${{\text{a}}_{\text{n}}}$
The formula for the arithmetic progression is given by,
\[{{\text{a}}_{\text{n}}}{\text{ = }}{\text{a}}{\text{ + }}\left( {{\text{n - 1}}} \right){\text{d}}\]
Substituting the values in the above equation we get,
\[
  {a_n} = 3 + \left( {100 - 1} \right) \times 2 \\
   = 3 + 198 \\
  {a_n} = 201 \;
 \]
Hence, the series $ = 3,5,7,9,11,........197,199,201$
Series with similar terms are $5,11,17,...........,197$
Here, the first term ${a_1} = 5$ and the common difference ${\text{d}}{\text{ = }}11{\text{ - }}5{\text{ = }}6$
We know that the last term,${{\text{a}}_{\text{n}}} = 197$
We are supposed to find the number of identical terms, i.e.; $n$
The formula for the arithmetic progression is given by,
\[{{\text{a}}_{\text{n}}}{\text{ = }}{\text{a}}{\text{ + }}\left( {{\text{n - 1}}} \right){\text{d}}\]
Substituting the values in the above equation we get,
\[
  {\text{197}}{\text{ = }}{\text{5}}{\text{ + }}\left( {{\text{n - 1}}} \right){\text{6}} \\
  {\text{197}}{\text{ - 5}}{\text{ = }}{\text{6n}}{\text{ - 6}} \\
  {\text{197}}{\text{ - 5}}{\text{ + 6}}{\text{ = }}{\text{6n}} \\
  {\text{198}}{\text{ = }}{\text{6n}} \\
  {\text{n}}{\text{ = }}\dfrac{{{\text{198}}}}{{\text{6}}} \\
  {\text{n}}{\text{ = }}{\text{33}} \;
 \]
Therefore, the number of identical terms in the sequence $2,5,8,11,$ ___ upto 100 terms and $3,5,7,9,11$____ upto $100$ terms are 33. Hence, option (B) is the correct answer.
So, the correct answer is “Option B”.

Note: Any intersection of two doubly infinite arithmetic progressions is either empty or another arithmetic progression, which can be determined by the Chinese remainder theorem. There exists a number common to all of the progressions in a family of doubly infinite arithmetic progressions if each pair of them has a non-empty intersection; that is, infinite arithmetic progressions form a Helly family.