Sum of first n natural numbers

(S_{58}=\frac{58\cdot59}{2}=1711). For a list starting from (1), use the natural-number sum formula.

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Here, \(n=65\), so \(\frac{65\times66}{2}=65\times33=2145\). Therefore, the correct answer is 2145. A value such as 2112 can result from using 65 or 66 incorrectly in the calculation. Exam tip: for a sum ending at \(n\), use \(\frac{n(n+1)}{2}\).

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Here, \(n=70\), so \(1+2+\cdots+70=\frac{70\times71}{2}=35\times71=2485\). Therefore, option C is correct. A value such as 2475 can result from a small multiplication or division error. Exam tip: divide the even factor by 2 first to calculate quickly.

The sum of the first n natural numbers is \(S_n=\frac{n(n+1)}{2}\). Here, \(n=80\), so \(S_{80}=\frac{80\times81}{2}=40\times81=3240\). Therefore, 3240 is the correct option. 3200 can result from the common error of using \(80\times80/2\); the formula must include \(n+1\). Exam tip: Substitute the last number for n and use \(n(n+1)/2\).

The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Hence, \(S_{72}=\frac{72\times73}{2}=36\times73=2628\). Therefore, 2628 is correct. A value such as 2618 can result from an arithmetic error in multiplication or addition. Exam tip: when \(n\) is even, divide \(n\) by 2 first to simplify the calculation.

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Substituting \(n=85\), we get \(\frac{85\times86}{2}=85\times43=3655\). Therefore, option C is correct. A value such as \(3645\) results from an arithmetic error in the calculation. Exam tip: use \(\frac{n(n+1)}{2}\) for the sum of consecutive natural numbers starting from 1.

The first n natural numbers run from 1 to n, so n is the largest number and \(n+1\) is the next natural number. Option B represents only the largest number, n. Exam tip: identify what each factor means before using a formula.

\(n\) and \(n+1\) are consecutive integers, so one of them must be even. Hence \(n(n+1)\) is always even, making division by 2 valid. Exam tip: consecutive integers always alternate between even and odd.

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Substituting \(n=95\), we get \(\frac{95\times96}{2}=95\times48=4560\). Therefore, option C is correct. A value such as \(4550\) may result from an error while multiplying the middle terms. Exam tip: cancel 2 with the even factor before multiplying.

The sum is \(S_n=\frac{n(n+1)}{2}\). Writing it as \(\frac{n^2+n}{2}\) shows that the highest power of n is 2, so it is a quadratic function. A linear function has highest power 1. Exam tip: expand the formula and check the highest power.

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Here, \(n=95\), so \(1+2+\cdots+95=\frac{95\times96}{2}=95\times48=4560\). A value such as 4655 can result from an incorrect multiplication or addition. Exam tip: whenever the sum from \(1\) to \(n\) is asked, apply \(\frac{n(n+1)}{2}\) directly.

(S_{48}=\frac{48\times49}{2}=1176). Divide the even number by (2) first to make calculation easier.

(S_{28}=\frac{28\times29}{2}=406), so (n=28). You can quickly check by substituting the options in the formula.

The sum is (S_{35}-S_{20}=630-210=420). When the sequence does not start from (1), take the difference of two sums.

The numbers of questions solved each day are 1, 2, 3, ..., 24. Therefore, the required total is the sum of the first 24 natural numbers: \(S=\frac{24\times25}{2}=300\). Hence, 300 is correct. Values such as 288 or 324 can result from an incorrect calculation of the sum formula. 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 \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{18}=\frac{18\times19}{2}=171\) and \(S_{12}=\frac{12\times13}{2}=78\). Therefore, \(S_{18}+S_{12}=171+78=249\). Option 239 is incorrect because it is not the sum of these two correct values. Exam tip: In such questions, first use \(S_n=\frac{n(n+1)}{2}\) to find each required sum separately.

The sum of the first n natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{25}=\frac{25\times26}{2}=325\) and \(S_{10}=\frac{10\times11}{2}=55\). Therefore, \(S_{25}-S_{10}=325-55=270\). This is also the sum of the natural numbers from 11 to 25. Getting 260 usually results from an error while calculating one of the two sums. Exam tip: the difference of two consecutive-range sums directly gives the sum of the terms between them.

(S_{34}=\frac{34\times35}{2}=595), so (n=34). Match the given sum with nearby triangular numbers.

(S_{30}=465) and (S_{18}=171), so the difference is (294). Write both sums correctly before subtracting.

Total chairs are (S_{32}=\frac{32\times33}{2}=528). A pattern increasing by row number forms (S_n).