Question

The number of common terms of the sequence $17,21,25,...,417$ and $16,21,26,...,466$ is
1. $21$
2. $19$
3. $20$
4. None of these

Answer 1

In a sequence of numbers the first term is referred to as $a$, the common difference referred as $d$, and the last term is referred to as $l$. Here we asked to find the common terms from the two sequences. We first need to find the total number of terms in both the sequences. Then we will equate the general ${n^{th}}$ terms of both sequences to find the common terms.

Formula used:
In a sequence, if the first term is $a$, the common difference is $d$, and the last term is $l$ then the total number in the sequence can be given as $n = \dfrac{{l - a}}{d} + 1$.
Then the ${n^{th}}$ term can be given as $a + (n - 1)d$.

Complete step by step answer:
The given two sequences are, $17,21,25,...,417$and$16,21,26,...,466$.
Let us first find the total number of terms for both sequences.
The first sequence:
The given sequence is $17,21,25,...,417$. Here the first term $a = 17$, the common difference$d = 21 - 17 = 4$, and the last term$l = 417$.
Thus, the total number of terms in this sequence is $n = \dfrac{{l - a}}{d} + 1$.
$\Rightarrow \dfrac{{417 - 17}}{4} + 1$
On simplifying this, we get
$n = \dfrac{{400}}{4} + 1$
On further simplification, we get
$n = 100 + 1 = 101$
Therefore, the total number of terms in the first sequence $n = 101$.
The second sequence:
The given sequence is $16,21,26,...,466$. Here the first term $a = 16$, the common difference$d = 21 - 16 = 5$, and the last term $l = 466$.
Thus, the total number of terms in this sequence is $n = \dfrac{{l - a}}{d} + 1$.
$\Rightarrow \dfrac{{466 - 16}}{5} + 1$
On simplifying this, we get
$n = \dfrac{{450}}{5} + 1$
On further simplification, we get
$n = 90 + 1 = 91$
Therefore, the total number of terms in the first sequence $n = 91$.
Now let us assume that the first sequence has a $n$ number of terms and the second sequence has a$m$ number of terms.
Now the ${n^{th}}$term of the first sequence is given by the formula $a + (n - 1)d$. Now substitute the values in $a + (n - 1)d$
$\Rightarrow 17 + (n - 1)4$
On simplifying this, we get$4n + 13$.
Then the ${m^{th}}$ term of the second sequence is given by the formula $a + (m - 1)d$. Now substitute the values in $a + (m - 1)d$
$\Rightarrow 16 + (n - 1)5$
On simplifying this, we get $5m + 11$
Then the common terms in both the sequence will be $4n + 13 = 5m + 11$
On simplifying the above expression, we get
$4n + 13 = 5m + 11$
$\Rightarrow 5m - 4n = 2$
Let us find the same possible values of the terms $m\& n$ that satisfy the above equation are$2,7,12,...,97$.
(That is, let $m\& n$ be two. Then, $5(2) - 4(4) = 2$
Let $m\& n$ be seven. Then, $5(7) - 4(7) = 7$and so on)
Now let us find the total number of terms in the sequence $2,7,12,...,97$.
Here we have a = 2$$$$d = 7 = 2 = 5$$$$\& $$$$l = 97. Then the total number of terms in that sequence is$n = \dfrac{{l - a}}{d} + 1 = \dfrac{{97 - 2}}{5} + 1$.
On simplifying this, we get
$n = \dfrac{{95}}{5} + 1$
On further simplification, we get
$n = 19 + 1 = 20$
Thus, the total number of terms in the sequence $2,7,12,...,97$is $20$
Let us see the options, option (1) $21$ is not the correct answer since we got the answer $20$ in our calculation.
Option (2) $19$ is not the correct answer since we got the answer $20$ in our calculation.
Option (3) $20$ is the correct answer as we got the same answer in our calculation.
Option (4) None of these is an incorrect answer since we got option (3) as a correct answer.
Hence, option (3) $20$ is the correct option.

Note: Here we are supposed to find the common term in both the sequence thus, we have equated {n^{th}}$$$$\& $$$${m^{th}} terms so that it will be the same and equal. Using the formula that we can find the ${n^{th}}$ term of both the sequences (general form).