Sum of first n natural numbers

Correct answer: C. (1)

(S_{77}=S_{76}+77) and (S_{75}=S_{76}-76), so the value is (1). Use differences of consecutive sums.

Correct answer: C. 3060

The sum of the first n natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{90}=\frac{90\times91}{2}=4095\) and \(S_{45}=\frac{45\times46}{2}=1035\). Therefore, the required difference is \(4095-1035=3060\). The value 4095 is only the sum of the first 90 natural numbers, not the difference. Exam tip: write \(S_{90}-S_{45}\) before calculating in such questions.

Correct answer: A. (118)

(S_{117}=6903) and (S_{118}=7021), so it first exceeds (7000) at (n=118). In boundary questions, check two nearby values.

Correct answer: B. \(219:25\)

The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{72}=\frac{72\times73}{2}=2628\) and \(S_{24}=\frac{24\times25}{2}=300\). Therefore, \(S_{72}:S_{24}=2628:300=219:25\), so option B is correct. Option \(219:20\) does not result from simplifying the ratio correctly. Exam tip: substitute the value of \(n\) carefully in the sum formula before reducing the ratio.

Correct answer: D. (66)

Total tiles are (S_r=2211), and (S_{66}=2211), so there are (66) rows. Such figures form a triangular number.

Correct answer: A. \(14\)

The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). From \(\frac{a(a+1)}{2}=2485\), we get \(a=70\), since \(\frac{70\times71}{2}=2485\). Similarly, \(\frac{b(b+1)}{2}=3570\) gives \(b=84\), since \(\frac{84\times85}{2}=3570\). Therefore, \(b-a=84-70=14\). A close distractor such as \(12\) may result from incorrectly identifying the indices. Exam tip: first express each given sum using \(\frac{n(n+1)}{2}\), then determine \(n\).

Correct answer: B. 4675

There are \(112-58+1=55\) terms from 58 to 112. Using the arithmetic progression sum formula, \(S=\frac{n}{2}(a+l)\), we get \(S=\frac{55}{2}(58+112)=\frac{55}{2}\times170=4675\). Hence, 4675 is correct. A value such as 4735 can result from an error in counting terms or finding the average. Exam tip: always include \(+1\) when counting terms from one endpoint to another.

Correct answer: C. 74

Let S_m be the sum of the first m natural numbers. In S_{m+2}-S_m, only the next two numbers, (m+1) and (m+2), remain. Thus, S_{m+2}-S_m=(m+1)+(m+2)=2m+3. Hence 2m+3=151, so 2m=148 and m=74. If m were 73, the difference would be 149, not 151. Exam tip: In differences of such sums, cancel the common initial terms and add only the extra terms.

Correct answer: D. 3450

The sum of the first 150 natural numbers is \(\frac{150\times151}{2}=11325\), and the sum of the first 125 natural numbers is \(\frac{125\times126}{2}=7875\). Therefore, the difference is \(11325-7875=3450\). It is also the sum of the numbers from 126 to 150. Although 3425 is close, it does not result from the correct calculation. Exam tip: interpret such a difference as the sum of the remaining consecutive terms.

Correct answer: A. (141)

(S_{140}=9870) and (S_{141}=10011), so it first exceeds (10000) at (n=141). In boundary questions, check the previous and next sums.

Correct answer: B. (5020)

(S_{35}=630), (S_{55}=1540), and (S_{75}=2850), so the total is (5020). Write all three sums separately and add.

Correct answer: A. \(S_n-S_{n-1}=n\)

\(S_n=1+2+\cdots+(n-1)+n\), whereas \(S_{n-1}=1+2+\cdots+(n-1)\). On subtraction, all common terms cancel and only \(n\) remains, so A is correct. Option B misses the last term by one. Exam tip: the difference of consecutive partial sums is the newly added term.

Correct answer: D. 5886

The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Therefore, \(S_{108}=\frac{108\times109}{2}=54\times109=5886\). Hence, 5886 is correct. A value such as 5866 may result from a small multiplication or addition error, but applying the formula gives 5886. Exam tip: When \(n\) is even, divide \(n\) by 2 before multiplying.

Correct answer: A. (2875)

(S_{64}=2080), (S_{84}=3570), and (S_{74}=2775), so the value is (2875). In mixed sums, write each (S_n) separately.

Correct answer: B. 20

The sum of the first n natural numbers is \(S_n=\frac{n(n+1)}{2}\). From \(S_{5p}=5050\), we get \(\frac{5p(5p+1)}{2}=5050\). Since \(\frac{100\times101}{2}=5050\), \(5p=100\), so \(p=20\). If p were 18, the index would be 90, whose sum is not 5050. Exam tip: recognise 5050 as half the product of two consecutive numbers first.

Correct answer: B. (104)

(S_{104}=\frac{104\times105}{2}=5460), so (n=104). Match (2S_n) with the product of consecutive numbers.

Correct answer: D. 504

Since \(S_{75}=S_{68}+(69+70+\cdots+75)\), the required number is the sum of the terms from 69 to 75. There are 7 terms, so their sum is \(\frac{7}{2}(69+75)=\frac{7}{2}\times144=504\). Hence, 504 is correct. Exam tip: When finding \(S_b-S_a\), add the terms from \(a+1\) to \(b\).

Correct answer: B. (582)

(S_{96}=4656), and (12.5%=\frac{1}{8}), so the value is (582). Converting percent to a fraction makes it easier.

Correct answer: C. (2626)

(S_{70}=2485), (S_{71}=2556), and (S_{69}=2415), so the value is (2626). Subtract carefully with nearby indices.

Correct answer: B. 15400

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Substituting \(n=175\), we get \(\frac{175\times176}{2}=175\times88=15400\). Therefore, option B is correct. Option A can result from an error in multiplication or halving. Exam tip: always remember to use \(n+1\) in the formula.

Correct answer: C. 1410

Here, \(S_n\) denotes the sum of the first \(n\) natural numbers, so \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{140}=9870\), \(S_{130}=8515\), and \(S_{10}=55\). Therefore, \(S_{140}-S_{130}+S_{10}=9870-8515+55=1410\). Note that \(S_{140}-S_{130}\) is the sum of the numbers from 131 to 140; adding \(S_{10}\) gives 1410. Exam tip: You can also verify \(S_a-S_b\) by summing the terms from \(b+1\) to \(a\).

Correct answer: C. 29

The sum of the first n natural numbers is \(S_n=\frac{n(n+1)}{2}\). Here, \(\frac{n(n+1)}{2}=1770\) gives \(n=59\), since \(\frac{59\times60}{2}=1770\). Therefore, \(2k+1=59\), so \(2k=58\) and \(k=29\). Option 30 may result from incorrectly taking the index as 60 instead of 59. Exam tip: first find n from \(S_n\), then equate the given index \(2k+1\) to n.

Correct answer: B. (200)

(S_{199}=19900) and (S_{200}=20100), so it first exceeds (20000) at (n=200). Checking nearby values is the safest method.

Correct answer: B. \(8S_n+1\) is a perfect square

Using \(S_n=\frac{n(n+1)}{2}\), we get \(8S_n+1=4n(n+1)+1=(2n+1)^2\), a perfect square. However, \(S_n\) itself is not always odd or a square. Exam tip: substitute the standard sum formula to test such identities.

Correct answer: A. 3123

Here, \(S_n\) denotes the sum of the first \(n\) natural numbers, so \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{36}=666\), \(S_{72}=2628\), and \(S_{18}=171\). Therefore, \(S_{36}+S_{72}-S_{18}=666+2628-171=3123\). The option 3143 can result from an error of 20 while subtracting. Exam tip: calculate each \(S_n\) separately before performing the final addition and subtraction.