Sum of first n natural numbers

The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Therefore, \(S_{28}=\frac{28\times29}{2}=14\times29=406\). Hence, 406 is correct. A value such as 416 usually results from an error in multiplication or division. Exam tip: Cancel the even factor with 2 first to calculate quickly.

The total is the sum of numbers from 1 to 20. Using \(n(n+1)/2\), we get \(20\times21/2=210\). The value 200 comes from incorrectly taking \(20\times10\). Exam tip: identify \(n\) before applying the formula.

The first natural number is 1. Therefore, the sum of the first 1 natural number is simply 1. Using \(\frac{n(n+1)}{2}\) with \(n=1\), we get \(\frac{1\times2}{2}=1\). Option A is not correct because 0 is not included here as the first natural number. Exam tip: “First \(n\)” means the sum of numbers from 1 to \(n\).

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Here, \(n=75\), so \(\frac{75\times 76}{2}=75\times 38=2850\). Therefore, 2850 is correct. A value such as 2825 does not result from multiplying 75 by its next number, 76, as required by the formula. Exam tip: For the sum from \(1\) to \(n\), directly use \(\frac{n(n+1)}{2}\).

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Thus, \(\frac{n(n+1)}{2}=36\). On substituting \(n=8\), we get \(\frac{8\times9}{2}=36\), so 8 is correct. For \(n=7\), the sum is only \(28\). Exam tip: for small answer choices, substitute each value in \(n(n+1)/2\) to check quickly.

The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Therefore, \(S_{23}=\frac{23\times24}{2}=23\times12=276\). Hence, option B is correct. \(264\) may result from incorrectly halving 24. Exam tip: do not forget to use \(n+1\) in the formula.

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Substituting \(n=27\), we get \(\frac{27\times28}{2}=27\times14=378\). Therefore, 378 is correct. Values such as 368 or 388 can result from an error while applying or calculating the formula. Exam tip: For the sum of the first \(n\) natural numbers, directly use \(\frac{n(n+1)}{2}\).

The sum of the first 15 natural numbers is found using \(S_n=\frac{n(n+1)}{2}\). Here, \(n=15\), so \(S_{15}=\frac{15\times16}{2}=120\). Therefore, 120 is the correct answer. 125 may result from an arithmetic error, but applying the formula gives 120. Exam tip: For the sum from 1 to \(n\), use \(\frac{n(n+1)}{2}\).

The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Therefore, \(S_{55}=\frac{55\times56}{2}=55\times28=1540\). Hence, option A is correct. \(1515\) is not the correct sum from \(1\) to \(55\). Exam tip: Cancel 2 with the even factor in \(n(n+1)\) before multiplying.

\(S_n=1+2+\cdots+n\), while \(S_{n-1}=1+2+\cdots+(n-1)\). On subtracting, all common terms cancel and only \(n\) remains. Exam tip: identify the last term added to the sum.

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Substituting \(n=36\), we get \(\frac{36\times37}{2}=18\times37=666\). Therefore, 666 is the correct answer. Although 656 is close, it is not obtained from the required formula. Exam tip: use \(\frac{n(n+1)}{2}\) for the sum of consecutive natural numbers from 1 to \(n\).

\(S_{15}=\frac{15\times16}{2}=120\). Cancelling by (2) before multiplication makes it easier.

This is the sum of the first (20) natural numbers, so (S_{20}=210). The dots (\cdots) show that the sequence continues.

The first 7 natural numbers are 1, 2, 3, 4, 5, 6 and 7. Their sum is \(1+2+3+4+5+6+7=28\). Therefore, 28 is correct. The value 21 is the sum of the first 6 natural numbers, so it is a close but incorrect option. Exam tip: the sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\).

(S_9=\frac{9\times10}{2}=45), so (n=9). In such questions, checking options in the formula is fast.

\(S_{12}=\frac{12\times13}{2}=78\). Dividing (12) by (2) first simplifies the calculation.

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Substituting \(n=18\) gives \(\frac{18\times19}{2}=171\), so option A is correct. \(190\) is the sum from \(1\) to \(19\), not from \(1\) to \(18\). Exam tip: in \(n(n+1)/2\), use \(n\) as the number of terms given in the question.

(S_{25}=\frac{25\times26}{2}=325). From (1) to (25) means the first (25) natural numbers.

The first 5 natural numbers are 1, 2, 3, 4, and 5. Their sum is \(1+2+3+4+5=15\), so option C is correct. The value 10 is the sum of only the first 4 natural numbers. Exam tip: the sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\).

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). So, \(\frac{n(n+1)}{2}=66\), giving \(n(n+1)=132\). Since \(11\times12=132\), \(n=11\). For \(n=12\), the sum is \(\frac{12\times13}{2}=78\), not 66. Exam tip: look for two consecutive numbers whose product equals \(2S\).