Find the sum of all multiple of 7 between 100 and 300
In this interval:
The first number divisible by 7 is 105
The last number divisile by 7 is 294
Each number is a term of AP:
an = a1 + ( n - 1) d
where
a1 = 105
an = 294
d = 7
an = 294 = 105 + ( n - 1) d =
105 + ( n - 1) ∙ 7 = 105 + 7 n - 7 = 98 + 7 n
294 = 98 + 7 n
294 - 98 = 7 n
196 = 7 n
n = 196 / 7 = 28
Sn is the sum of n terms in the arithmetic progression:
Sn = ( n / 2 ) ( a1 + an )
In tis case:
n = 28
Sn = S28
Sn = ( n / 2 ) ( a1 + an )
Sn = S28 = ( 28 / 2 ) ( 105 + 294 )
Sn = 14 ∙ 399
Sn = 5 586
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Well, let's get calculating!
To find the sum of all multiples of 7 between 100 and 300, we can use a little trick. The first multiple of 7 in that range is 105, and the last is 294.
Now, let's count the number of multiples of 7 between 105 and 294. We divide the range (294 - 105 = 189) by 7 and get 27.
So, we have 27 multiples of 7 between 100 and 300.
To find the sum, we use a formula—take the average of the first and the last number (105 + 294 = 399) and multiply it by the number of multiples (27). So, the sum is (399 * 27) = 10,773.
That's a big number, almost as big as the number of clowns that can fit into a tiny car!
To find the sum of all multiples of 7 between 100 and 300, follow these steps:
1. Determine the first multiple of 7 within the given range. Divide 100 by 7 and find the quotient: 100 ÷ 7 = 14 remainder 2. Since the remainder is 2, the first multiple of 7 within the range is 7 × 15 = 105.
2. Determine the last multiple of 7 within the given range. Divide 300 by 7 and find the quotient: 300 ÷ 7 = 42 remainder 6. Since the remainder is 6, the last multiple of 7 within the range is 7 × 42 = 294.
3. Calculate the number of terms in the sequence. To find the number of terms, subtract the first multiple from the last multiple and divide by the common difference (which is 7). In this case, (294 - 105) ÷ 7 = 189 ÷ 7 = 27. Therefore, there are 27 multiples of 7 between 100 and 300.
4. Use the formula for the sum of an arithmetic series to calculate the sum:
S = (n/2)(a + l)
where S represents the sum, n represents the number of terms (27 in this case), a represents the first term (105), and l represents the last term (294).
S = (27/2)(105 + 294) = (13.5)(399) = 5386.5
Hence, the sum of all multiples of 7 between 100 and 300 is 5386.5.