Sum of first n natural numbers

Correct answer: A. 253

The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Therefore, \(S_{22}=\frac{22\times23}{2}=11\times23=253\). The option 242 may result from calculating \(22\times11\), but the correct formula must include \(n+1=23\). Exam tip: Use \(\frac{n(n+1)}{2}\) and divide the even factor by 2 before multiplying.

Correct answer: A. \(\frac{n(n+1)}{2}\)

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). For \(n=4\), \(1+2+3+4=10\), and the formula gives \(\frac{4\times5}{2}=10\). Exam tip: include the \(n+1\) factor.

Correct answer: C. 300

The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Therefore, \(S_{24}=\frac{24\times25}{2}=12\times25=300\). Hence, 300 is correct. A value such as \(288\) may result from an incorrect multiplication or addition. Exam tip: Cancel the even factor with 2 first to calculate quickly.

Correct answer: C. 6

The first three natural numbers are 1, 2, and 3. Therefore, their sum is \(1+2+3=6\). Option 3 is only the third natural number, not the sum. Exam tip: remember that the sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\).

Correct answer: A. 1830

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Substituting \(n=60\), we get \(\frac{60\times61}{2}=30\times61=1830\). A value such as \(1800\) may result from incorrectly using \(60\times60/2\), but the formula must include \(n+1=61\). Exam tip: for consecutive natural numbers starting from 1, use \(\frac{n(n+1)}{2}\).

Correct answer: C. 20

Use the governing formula S = n(n + 1)/2 for the sum of the first n natural numbers. With S = 210, we have n(n + 1)/2 = 210, so n(n + 1) = 420. For n = 20, the product is 20 × 21 = 420, and the sum is 20 × 21 / 2 = 210. Thus option C is correct. Checking the nearby distractors confirms the result: n = 18 gives 171, n = 19 gives 190, and n = 21 gives 231. Since n represents a count, the positive integer solution is required. Algebraically, n² + n − 420 = 0 factors as (n − 20)(n + 21) = 0; the admissible root is n = 20.

Correct answer: C. 153

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Substituting \(n=17\), we get \(\frac{17\times18}{2}=17\times9=153\), so 153 is correct. The value 170 may result from using \(17\times10\), but that is not the correct sum formula. Exam tip: for the sum of the first \(n\) natural numbers, use \(\frac{n(n+1)}{2}\).

Correct answer: B. 190

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Here, \(n=19\), so \(\frac{19\times20}{2}=19\times10=190\). The values 180, 200, and 210 do not result from this formula. Exam tip: use the last number in the series as \(n\) before applying the formula.

Correct answer: C. 3

The first 2 natural numbers are 1 and 2. Their sum is \(1+2=3\), so option C is correct. Option 2 is only the second natural number, not the sum. Exam tip: the sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\).

Correct answer: A. \(\frac{n(n+1)}{2}\)

The natural numbers from 1 to n have sum \(\frac{n(n+1)}{2}\). The expression \(\frac{n(n-1)}{2}\) corresponds to the sum up to \(n-1\). Exam tip: substitute n=1; a correct formula must give 1.

Correct answer: A. जोड़ी जाने वाली प्राकृतिक संख्याओं की कुल संख्या

Here, \(n\) is the number of consecutive natural numbers being added. For example, in the sum from 1 to 5, \(n=5\). Although 5 is also the largest term, \(n\) denotes the number of terms. Exam tip: identify the term count before using the formula.

Correct answer: C. 630

The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Therefore, \(S_{35}=\frac{35\times36}{2}=35\times18=630\). Hence, 630 is correct. Values such as 620 and 640 do not result from this formula. Exam tip: Cancel 2 with the even factor in \(n(n+1)\) before multiplying to calculate quickly.

Correct answer: A. 55

The sum of the first 10 natural numbers is found using \(S_n = \frac{n(n+1)}{2}\). Here, \(n=10\), so \(S_{10}=\frac{10\times11}{2}=55\). Therefore, 55 is correct. The value 45 is the sum from 1 to 9, so it is a close but incorrect option. Exam tip: For the sum from 1 to \(n\), use \(\frac{n(n+1)}{2}\) directly.

Correct answer: B. 1035

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Substituting \(n=45\), we get \(\frac{45\times46}{2}=45\times23=1035\). Therefore, 1035 is the correct answer. A value such as 1045 may result from an incorrect calculation using the formula. Exam tip: For the sum of consecutive natural numbers, use \(\frac{n(n+1)}{2}\).

Correct answer: C. 25

The sum of the first n natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(\frac{n(n+1)}{2}=325\), so \(n(n+1)=650\). Since \(25\times26=650\), \(n=25\). For \(n=24\), the sum is \(300\), so it is a close but incorrect option. Exam tip: look for consecutive factors of 650 in such questions.

Correct answer: A. 406

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.

Correct answer: B. 210

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.

Correct answer: B. 1

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 \(n\).

Correct answer: C. 2850

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 \(\frac{n(n+1)}{2}\).

Correct answer: C. 8

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.

Correct answer: B. 276

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.

Correct answer: B. 378

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 \(\frac{n(n+1)}{2}\).

Correct answer: C. 120

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 \(\frac{n(n+1)}{2}\).

Correct answer: A. 1540

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.

Correct answer: A. \(n\)वीं प्राकृतिक संख्या

\(S_n=1+2+\cdots+n\), 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.