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MICAT Sequences and Series — practice questions

Explore 15 free Sequences and Series questions in MICAT Quantitative Aptitude. Open each question or exercise to read its content, reference answers and available explanation.

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Sequences and Series questions and exercises with answers

Showing questions 1–15 of 15.

The $n^\text{th}$ term of an Arithmetic Progression (AP) with first term $a$ and common difference $d$ isRead the answer and worked solution →The sum of the first $n$ terms of an AP with first term $a$ and last term $l$ isRead the answer and worked solution →The 10th term of an AP with first term $a = 5$ and common difference $d = 3$ isRead the answer and worked solution →The $n^\text{th}$ term of a Geometric Progression (GP) with first term $a$ and common ratio $r$ isRead the answer and worked solution →The sum of the first $n$ terms of a GP with first term $a$ and common ratio $r \neq 1$ isRead the answer and worked solution →The sum to INFINITY of the GP $1 + 1/2 + 1/4 + 1/8 + \ldots$ isRead the answer and worked solution →The Arithmetic Mean (AM) of two positive numbers $a$ and $b$ isRead the answer and worked solution →The Geometric Mean (GM) of $4$ and $9$ isRead the answer and worked solution →Sum of the first $n$ natural numbers, $1 + 2 + 3 + \ldots + n$, equalsRead the answer and worked solution →Sum of the SQUARES of the first $n$ natural numbers, $1^2 + 2^2 + 3^2 + \ldots + n^2$, equalsRead the answer and worked solution →Sum of the CUBES of the first $n$ natural numbers, $1^3 + 2^3 + \ldots + n^3$, equalsRead the answer and worked solution →Three numbers are in AP. Their sum is $24$ and their product is $440$. Find the smallest of the three.Read the answer and worked solution →If the AM and GM of two positive numbers are $5$ and $4$ respectively, the numbers areRead the answer and worked solution →The AM-GM INEQUALITY for two POSITIVE numbers $a$ and $b$ states thatRead the answer and worked solution →GMAT · Quantitativeaptitude · Coordinate Geometry — Memory FlipRead the answer and worked solution →

Worked practice questions

The $n^\text{th}$ term of an Arithmetic Progression (AP) with first term $a$ and common difference $d$ is

  1. $a_n = a + (n-1)d$
  2. $a_n = a + nd$
  3. $a_n = an + d$
  4. $a_n = a + n/d$
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Correct answer: A. $a_n = a + (n-1)d$

1. Your syllabus: each term is obtained by adding the common difference $d$ to the previous term. 2. Terms: $a_1 = a$, $a_2 = a + d$, $a_3 = a + 2d$, $\ldots$ 3. Pattern: the $n^\text{th}$ term has $(n-1)$ additions of $d$, so $a_n = a + (n-1)d$. 4. Example: AP $3, 7, 11, 15, \ldots$ with $a = 3$ and $d = 4$. The 5th term: $a_5 = 3 + (5-1)(4) = 3 + 16 = 19$. ✓ 5. Options B and C have wrong functional forms. Option D divides instead of multiplying. _Source: syllabus Mathematics, Ch 8 "Sequences and Series", §8.2 (Arithmetic Progression), p. 2–3._

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The sum of the first $n$ terms of an AP with first term $a$ and last term $l$ is

  1. $S_n = n(a + l)$
  2. $S_n = (n/2)(a + l)$
  3. $S_n = n^2(a + l)/2$
  4. $S_n = a + l$
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Correct answer: B. $S_n = (n/2)(a + l)$

1. syllabus derives the AP sum. 2. Pair up terms: 1st + last = $a + l$; 2nd + second-last = $a + l$; etc. Each pair sums to $a + l$. 3. With $n$ terms, there are $n/2$ such pairs (works whether $n$ is even or odd by symmetry). 4. Sum: $S_n = (n/2)(a + l)$. 5. Equivalently, using $l = a + (n-1)d$: $S_n = (n/2)\,[2a + (n-1)d]$. 6. Example: $1 + 2 + 3 + \ldots + 100 = (100/2)(1 + 100) = 50 \cdot 101 = 5050$ ✓ (Gauss's famous childhood calculation). 7. Other options have wrong dimensions or factors. _Source: syllabus Mathematics, Ch 8, §8.2 (AP sum formula), p. 3–4._

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The 10th term of an AP with first term $a = 5$ and common difference $d = 3$ is

  1. $30$
  2. $33$
  3. $35$
  4. $32$
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Correct answer: D. $32$

1. AP formula: $a_n = a + (n-1)d$. 2. Substitute $a = 5$, $d = 3$, $n = 10$. 3. $a_{10} = 5 + (10-1)(3) = 5 + 9 \cdot 3 = 5 + 27 = 32$. 4. Quick verification by counting: starts at 5, add 3 nine times → 5+3=8, 11, 14, 17, 20, 23, 26, 29, 32. ✓ 5. Option A uses $n$ instead of $(n-1)$. Option C overshoots by 1. Option D adds 30 instead of 27. _Source: syllabus Mathematics, Ch 8, §8.2 + Examples, p. 3–4._

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