Sum of first n natural numbers

Given \(S_n=\frac{n(n+1)}{2}\). Substituting \(n=6\), \(S_6=\frac{6(6+1)}{2}=\frac{6\times7}{2}=21\). Therefore, the correct answer is 21. Getting 18 usually results from an error in multiplication or division by 2. Exam tip: first find \(n+1\), then multiply and divide by 2.

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Substituting \(n=30\) gives \(\frac{30\times31}{2}=15\times31=465\). A value such as \(435\) can result from incorrectly using \(30\times29/2\). Exam tip: for the sum of natural numbers, always use \(n+1\) in the formula.

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Here, \(n=14\), so \(\frac{14\times15}{2}=105\). Therefore, the correct answer is 105. The option 110 may result from an incorrect calculation of \(14\times15\div2\). Exam tip: use the last number as \(n\) in the formula.

The first 9 natural numbers are \(1,2,3,\ldots,9\). The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Substituting \(n=9\) gives \(\frac{9\times10}{2}=45\). The close distractor 36 is the sum of the first 8 natural numbers. Exam tip: remember that the formula uses both \(n\) and \(n+1\).

The sum of the first (n) natural numbers is (S_n=\frac{n(n+1)}{2}). This formula is the key formula of this topic.

For \(S_n=1+2+\cdots+n\), the correct sum formula is \(\frac{n(n+1)}{2}\). The expression \(\frac{n(n-1)}{2}\) corresponds to the sum up to \(n-1\). Exam tip: check that the factor is \(n+1\), not \(n-1\).

The total is the sum of the dots from the first to the 11th row: \(1+2+3+\cdots+11\). The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Therefore, \(\frac{11\times12}{2}=66\), so 66 is correct. Note that 55 is the sum of only the first 10 natural numbers. Exam tip: For this row pattern, use \(\frac{n(n+1)}{2}\).

The daily readings form the natural-number sequence 1, 2, 3, ..., 16. Therefore the total is the sum of the first 16 natural numbers, not the product of the number of days and a constant daily amount. Use \(S_n=\frac{n(n+1)}{2}\). With n=16, \(S_{16}=\frac{16\times17}{2}=8\times17=136\) pages.

Thus option C is correct. Another way to check is to pair the first and last readings: 1+16=17, 2+15=17, and so on. There are 8 such pairs, giving \(8\times17=136\). The sequence increases by one page each day, so answers such as 126 or 144 do not match the required arithmetic-series total. The supplied answer and explanation are accurate.

The first 4 natural numbers are 1, 2, 3, and 4. Therefore, their sum is 1 + 2 + 3 + 4 = 10. The value 9 is only the sum of 2 + 3 + 4, so it is not correct. Exam tip: The sum of the first n natural numbers is n(n+1)/2.

(S_{13}=\frac{13\times14}{2}=91), so (n=13). Check nearby sums to identify the correct (n).

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Here \(n=19\), so the sum is \(\frac{19\times20}{2}=190\). Hence, option C is correct. The value 180 may result from an incorrect multiplication of 19 and 20. Exam tip: In the formula, substitute the last natural number for \(n\).

The first (2) natural numbers are (1) and (2), so the sum is (3). Listing terms is safe in basic questions.

The number of bricks in the layers runs from 1 to 10, so the total is \(1+2+3+\cdots+10\). The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Thus, \(\frac{10\times11}{2}=55\), so 55 is correct. The value 45 is the sum from 1 to 9, making it a close but incorrect option. Exam tip: for the sum from 1 to \(n\), use \(\frac{n(n+1)}{2}\) directly.

\(S_{22}=\frac{22\times23}{2}=253\). Halve the even number (22) and calculate \(11\times23\).

The sum of the first n natural numbers is \(S_n=\frac{n(n+1)}{2}\), so A is correct. \(\frac{n(n-1)}{2}\) gives the sum only up to \(n-1\). Exam tip: test any formula with \(n=1\).

The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Therefore, \(S_{17}=\frac{17\times18}{2}=17\times9=153\). Hence, option C is correct. A value such as \(148\) does not result from the correct calculation of \(\frac{17\times18}{2}\). Exam tip: For the sum of the first \(n\) natural numbers, directly use \(\frac{n(n+1)}{2}\).

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Here, \(n=50\), so \(\frac{50\times51}{2}=25\times51=1275\). Therefore, the correct answer is 1275. Note that 1225 equals \(\frac{49\times50}{2}\), the sum from 1 to 49. Exam tip: For consecutive natural numbers starting from 1, use \(n(n+1)/2\).

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Substituting \(n=24\), we get \(\frac{24\times25}{2}=12\times25=300\). Therefore, 300 is the correct option. Although 288 is close, it is not obtained from the formula. Exam tip: In such questions, use \(n(n+1)/2\) and divide the even factor by 2 first.

(S_{15}=120) because (\frac{15\times16}{2}=120). In such questions, verify the answer back in the formula.

The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Therefore, \(S_{40}=\frac{40\times41}{2}=20\times41=820\). Hence, 820 is correct. An answer such as 800 may result from incorrectly using 40 in place of \(n+1\). Exam tip: Write \(n(n+1)/2\) before substituting the value of \(n\).