NIMCET 2013 — Mathematics PYQ

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Question

NIMCET 2013 — Mathematics PYQ

NIMCET | Mathematics | 2013

The sum of the integers between 200 and 400, that are multiples of 7, is:

Choose the correct answer:

Explanation

Solution

To find the sum, we first identify the sequence of multiples of 7 between 200 and 400.

Step 1: Find the first and last terms

Dividing 200 by 7: $200 \div 7 = 28$ with a remainder of 4. The first multiple after 200 is $7 \times 29 = 203$ .
Dividing 400 by 7: $400 \div 7 = 57$ with a remainder of 1. The last multiple before 400 is $400 - 1 = 399$ .

So, the sequence is: 203, 210, 217, ..., 399.

This is an Arithmetic Progression (AP) where:

First term ( $a$ ) = $203$
Last term ( $l$ ) = $399$
Common difference ( $d$ ) = $7$

Step 2: Find the number of terms ( $n$ )

Using the formula for the $n^{th}$ term:

l = a + (n - 1)d

399 = 203 + (n - 1)7

399 - 203 = (n - 1)7

196 = (n - 1)7

n - 1 = \frac{196}{7}

n - 1 = 28

n = 29

Step 3: Find the sum of the sequence ( $S_n$ )

Using the sum formula for an AP:

S_n = \frac{n}{2} (a + l)

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

S_{29} = \frac{29}{2} (602)

S_{29} = 29 \times 301

S_{29} = 8729

Final Answer: The sum is 8729.