Sum of first n natural numbers

Correct answer: C. 156

The sum of the first n natural numbers is \(S_n=\frac{n(n+1)}{2}\). Therefore, \(\frac{n(n+1)}{2}=12246\). On substituting \(n=156\), we get \(\frac{156\times157}{2}=78\times157=12246\). Hence, 156 is correct. For the closest distractor, \(n=155\) gives \(\frac{155\times156}{2}=12090\), not 12246. Exam tip: when answer choices are given, substituting them in the formula is often the quickest reliable check.

Correct answer: B. (125)

(S_{125}-S_{70}=7875-2485=5390), so none of the listed options matches (5250). Check the difference first.

Correct answer: B. 8775

Subtracting the sum from 1 to 150 from the sum from 1 to 200 leaves only the terms from 151 to 200. Thus, \(S_{200}-S_{150}=\frac{200\times201}{2}-\frac{150\times151}{2}=20100-11325=8775\). Therefore, option B is correct. A value such as 8825 can result from an arithmetic error in the addition or subtraction. Exam tip: recognise such a difference directly as the sum of the remaining consecutive terms, from 151 to 200.

Correct answer: A. (1572)

For the sum of the first \\(n\\) natural numbers, \\(S_n=\\frac{n(n+1)}2\\). We are told that \\(S_n=20100\\). Solving \\(\\frac{n(n+1)}2=20100\\) gives \\(n(n+1)=40200\\), and \\(200\\times201=40200\\), so \\(n=200\\). Therefore \\(S_n-S_{n-8}\\) is the sum of the last eight numbers through 200.

Those numbers are 193, 194, 195, 196, 197, 198, 199, and 200. Their sum can be found by pairing the first and last terms: there are 8 terms and each pair totals 393, giving \\(8\\times\\frac{193+200}{2}=8\\times196.5=1572\\). Hence option A is correct.

Correct answer: B. (19)

(1770=S_{59}), so (3n=59), which does not give an integer (n). The correct conclusion is no integer (n).

Correct answer: A. 10100

There are
\(150-50+1=101\) terms from 50 to 150, including both endpoints. Their average is
\(\frac{50+150}{2}=100\). Hence, the sum is
\(101\times100=10100\). The option 10050 results from an error in counting the inclusive terms or finding the average. Exam tip: for an inclusive range from \(a\) to \(b\), use \(b-a+1\) for the number of terms.

Correct answer: C. 6903

Here, \(S_n-S_{n-3}\) represents the sum of the three consecutive natural numbers \(n\), \(n-1\), and \(n-2\). Thus, \(S_n-S_{n-3}=n+(n-1)+(n-2)=3n-3\). From \(3n-3=348\), we get \(n=117\). Therefore, \(S_{117}=\frac{117\times118}{2}=6903\). Hence, option C is correct. The value 6786 may appear close to \(S_{116}\), but the required value is for \(n=117\). Exam tip: \(S_n-S_{n-k}\) is the sum of the last \(k\) terms.

Correct answer: B. 194:109

The sum of the first n natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{96}=\frac{96\times97}{2}=4656\) and \(S_{72}=\frac{72\times73}{2}=2628\). Dividing \(4656:2628\) by 24 gives \(194:109\). The ratio \(96:72\) compares only the numbers of terms, not the sums. Exam tip: when comparing such sums, include both factors \(n\) and \(n+1\).

Correct answer: C. (13545)

(15400=S_{175}), so (S_{165}=\frac{165\cdot166}{2}=13695). The listed options do not contain the correct value.

Correct answer: B. (5099.5)

Adding the two equations gives (2S_n=10199), so (S_n=5099.5). Such values also help check the consistency of the question.

Correct answer: A. (11275)

To add consecutive natural numbers from 201 to 250, use the sum formula for an initial segment and subtract the numbers up to 200. The formula is \\(S_n=\frac{n(n+1)}{2}\\). Subtracting \\(S_{200}\\) removes 1 through 200 and leaves exactly 201 through 250, so no endpoint is missed.

Calculate \\(S_{250}=\frac{250\times251}{2}=31375\\), and \\(S_{200}=\frac{200\times201}{2}=20100\\). Therefore, the required sum is \\(S_{250}-S_{200}=31375-20100=11275\\). A quick check uses 50 terms with average \\(\frac{201+250}{2}=225.5\\), giving \\(50\times225.5=11275\\). Hence option A is correct.

Correct answer: D. 120

Here, \(S_k\) denotes the sum of the first \(k\) natural numbers. Therefore, \(S_{n+1}-S_{n-4}\) leaves the five terms from \((n-3)\) to \((n+1)\): \[(n-3)+(n-2)+(n-1)+n+(n+1)=5n-5.\] Thus, \(5n-5=595\), so \(5n=600\) and \(n=120\). If \(n=119\), the sum would be \(590\), not \(595\). Exam tip: In \(S_a-S_b\), write the terms from \(b+1\) to \(a\).

Correct answer: A. 12195

The sum of the first n natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{180}=\frac{180\times181}{2}=16290\) and \(S_{90}=\frac{90\times91}{2}=4095\). Hence, the difference is \(16290-4095=12195\). It is also the sum of the integers from 91 to 180; a value such as 12295 does not result from the formula or subtraction. Exam tip: write such a difference as \(S_{180}-S_{90}\).

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

The natural numbers 1, 2, 3, …, n form an arithmetic progression. Thus, \(S_n=\frac{n}{2}[2+(n-1)] = \frac{n(n+1)}{2}\). Option B gives the sum up to \(n-1\). Exam tip: check that the final term is n before applying the formula.

Correct answer: C. (200)

(S_{200}-S_{100}=20100-5050=15050), so none of the listed options matches (15150). Always check the difference.

Correct answer: A. 2054

The sum of the first \(n\) natural numbers is \(S_n=\frac{n(n+1)}{2}\). Thus, \(S_{78}=\frac{78\times79}{2}=3081\), \(S_{52}=\frac{52\times53}{2}=1378\), and \(S_{26}=\frac{26\times27}{2}=351\). Hence, \(S_{78}-S_{52}+S_{26}=3081-1378+351=2054\). Exam tip: evaluate each \(S_n\) separately, then apply the signs in the expression carefully.

Correct answer: C. (4950)

(1225=S_{49}), so (2n=98) and (S_{98}=\frac{98\cdot99}{2}=4851). The correct option should be (B).

Correct answer: A. 120

Here, \(S_n\) denotes the sum of the first \(n\) natural numbers. Thus, \(S_{n+4}-S_n\) leaves only the next four terms: \((n+1)+(n+2)+(n+3)+(n+4)=4n+10\). Therefore, \(4n+10=490\), so \(4n=480\) and \(n=120\). For the closest distractor, \(n=121\) would give a difference of \(494\), not 490. Exam tip: Write \(S_{n+k}-S_n\) as the sum of terms from \(n+1\) to \(n+k\).

Correct answer: C. 7525

Let the sum of the first n natural numbers be \(S_n=\frac{n(n+1)}{2}\). Therefore, the required result is \(S_{175}-S_{125}=\frac{175\times176}{2}-\frac{125\times126}{2}=15400-7875=7525\). It is also the direct sum of the numbers from 126 to 175. An option such as 7475 results from an error in evaluating a sum or subtraction. Exam tip: the difference of two sums with the same starting term equals the sum of the remaining consecutive terms.

Correct answer: B. (121:81)

(S_{120}:S_{80}=7260:3240=121:54), so the listed options do not contain the correct simplified ratio. Simplify fully.

Correct answer: A. 885

The sum of the first n natural numbers is \(S_n=\frac{n(n+1)}{2}\). From \(\frac{n(n+1)}{2}=11325\), we get \(n=150\). Therefore, \(S_n-S_{n-6}\) is the sum of the last 6 numbers, \(145+146+147+148+149+150=885\). Option 875 is not the correct sum of these six terms. Exam tip: \(S_n-S_{n-r}\) always represents the sum of the last r terms up to n.

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

The sum \(1+2+\cdots+n\) is given by \(\frac{n(n+1)}{2}\). In contrast, \(\frac{n(n-1)}{2}\) represents the sum only up to \(n-1\). Exam tip: check that the formula contains the consecutive factors n and \(n+1\).

Correct answer: A. 9372

There are \(150-63+1=88\) terms from 63 to 150. Hence, using the sum of an arithmetic progression, \(\frac{88}{2}(63+150)=44\times213=9372\). Therefore, option A is correct. A nearby value such as 9369 can result from a small arithmetic error. Exam tip: add 1 while counting terms when both endpoints are included.

Correct answer: C. \(141\)

Here, \(S_n\) denotes the sum of the first \(n\) natural numbers. In \(S_n-S_{n-3}\), only the last three terms remain: \((n-2)+(n-1)+n\). Thus, \(3n-3=420\), so \(3n=423\) and \(n=141\). If \(n=140\), the difference would be \(417\), so it is not correct. Exam tip: In differences of such sums, cancel the common terms and write only the remaining terms.

Correct answer: B. 5220

The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). Thus, \(S_{144}=\frac{144\times145}{2}=10440\). Half of this sum is \(\frac{10440}{2}=5220\), so option B is correct. \(10440\) is the complete sum, not its half. Exam tip: After finding a sum, carefully apply any extra operation such as “half” or “double.”