Finding common terms in two arithmetic progressions

Finding common terms in two arithmetic progressions

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Finding common terms in two arithmetic progressions

The number of common terms in two A.P.’s 2,7,12,17 500 terms and 1,8,15,22 300 terms is

Find the number of terms common to the two arithmetic progression 5,9,13,17,…,216 and 3,9,15,21,…..,321

The first common term in two Aps is 9.
LCm of their common difference = LCM (4,6) =12
New AP of common elements is 9,21,33,45,….,
Let is contain n terms. Then,

T_{n} > 216 \Rightarrow 9 + (n - 1) \times 12 \leq 216 \Rightarrow (n - 1) \times 12 \geq 207

\Rightarrow (n - 1) \times 12 = 204 \Rightarrow n = 18

So, there are 18 common terms in two APs.

The number of terms common to the two arithmetic progressions
$3, 7, 11, \dots, 407$ and $2, 9, 16, \dots, 709$ is

Solution

Given A.P.s are

$3, 7, 11, 15, 19, 23 \dots, 407$ (common difference is $4$ )

$2, 9, 16, 23 \dots, 709$ (common difference is $7$ )

We can observe that the first common term is $23$ .

( $6^{t h}$ term in $1^{s t}$ A.P. and $4^{t h}$ term in $2^{n d}$ A.P. are same.)

Let the number of common terms be $^{'} n^{'}$ .

They will be in A.P. with common difference $= 28$ ( i.e. (LCM $(4, 7)$ )

Now, the last common term can’t be greater than $407$ . [Last term of the first A.P]

$\Rightarrow 23 + (n - 1) 28 \leq 407$

$\Rightarrow n \leq \frac{384}{28} + 1$

$\Rightarrow n \leq 14.7 \Rightarrow n = 14$