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Solution

Step 1: Find the first multiple of 7 after 200
$200 \div 7 = 28.57...$ so first multiple $= 29 \times 7 = 203$

Step 2: Find the last multiple of 7 before 400
$400 \div 7 = 57.14...$ so last multiple $= 57 \times 7 = 399$

Step 3: Identify the AP
First term $a = 203$
Last term $l = 399$
Common difference $d = 7$

Step 4: Find number of terms $n$
$l = a + (n-1)d$
$399 = 203 + (n-1) \times 7$
$399 - 203 = (n-1) \times 7$
$196 = (n-1) \times 7$
$n - 1 = 28$
$n = 29$

Step 5: Find the Sum
$S_n = \dfrac{n}{2}(a + l)$

$S_{29} = \dfrac{29}{2}(203 + 399)$

$S_{29} = \dfrac{29}{2} \times 602$

$S_{29} = 29 \times 301$

$S_{29} = 8729$

Answer: The sum of all multiples of 7 between 200 and 400 is $\boxed{8729}$