The number of terms common to the two A.P.s $2 + 5 + 8 + 11 + . . . . . . . + 98$ and $3 + 8 + 13 + 18 + . . . . + 198$ are

33

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40

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7

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None of these

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Solution

The correct option is D $7$
For first A.P
$2 + 5 + 8 + 11 + . . . . . . + 98$
$a = 2, a_{n} = 98, d = 3$
$a_{n} = a + (n - 1) d$
$98 = 2 + (n - 1) 3$
$98 = 2 + 3 n - 3$
$3 n = 99$
$n = 33$
Number of term $= 33$
For first A.P
$3 + 8 + 13 + 18 + . . . . . . + 198$
$a = 3, a_{n} = 198, d = 5$
$a_{n} = a + (n - 1) d$
$198 = 3 + (n - 1) 5$
$198 = 3 + 5 n - 5$
$5 n = 200$
$n = 40$
No of terms $= 40$
Common terms= $40 - 33 = 7$

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