CUET UG Mathematics Algebra Sequences and Series AP GP Special Series Sum to Infinity

Must‑Know

• An arithmetic progression (AP) is a sequence where each term after the first is obtained by adding a constant d (common difference) to the preceding term; example: 2, 5, 8, 11,… with d = 3.
• The nth term of an AP is given by $ a_n = a + (n - 1)d $, where a is the first term and d is the common difference; for a = 3, d = 4, $ a_5 = 3 + (5 - 1) \times 4 = 19 $.
• The sum of first n terms of an AP is $ S_n = \frac{n}{2}[2a + (n - 1)d] $ or $ S_n = \frac{n}{2}(a + l) $, where l is the last term; for a = 2, d = 3, $ S_4 = \frac{4}{2}[2(2) + 3(3)] = 2[4 + 9] = 26 $.
• In an AP, if mth and nth terms are equal, then $ a + (m - 1)d = a + (n - 1)d $ implies $ d(m - n) = 0 $, so either d = 0 or m = n.
• A geometric progression (GP) is a sequence where each term is obtained by multiplying the previous term by a constant r (common ratio); example: 3, 6, 12, 24,… with r = 2.
• The nth term of a GP is $ a_n = ar^{n-1} $, where a is the first term; for a = 5, r = 3, $ a_4 = 5 \times 3^3 = 135 $.
• The sum of first n terms of a GP is $ S_n = \frac{a(r^n - 1)}{r - 1} $ when $ r \ne 1 $; for a = 2, r = 3, $ S_3 = \frac{2(27 - 1)}{3 - 1} = \frac{52}{2} = 26 $.
• For a GP with $ |r| < 1 $, sum to infinity is $ S_\infty = \frac{a}{1 - r} $; for a = 1, r = 1/2, $ S_\infty = \frac{1}{1 - 0.5} = 2 $.
• If three numbers are in AP, they can be assumed as $ a - d, a, a + d $; their sum is 3a.
• If three numbers are in GP, they can be assumed as $ \frac{a}{r}, a, ar $; their product is $ a^3 $.
• The arithmetic mean (AM) between two numbers a and b is $ \frac{a + b}{2} $; AM between 4 and 10 is $ \frac{4 + 10}{2} = 7 $.
• The geometric mean (GM) between two numbers a and b is $ \sqrt{ab} $; GM between 4 and 9 is $ \sqrt{36} = 6 $.
• For any two positive numbers, $ AM \ge GM $; equality holds only when numbers are equal.
• The sum of first n natural numbers is $ \frac{n(n + 1)}{2} $; sum of first 10 natural numbers is $ \frac{10 \times 11}{2} = 55 $.
• The sum of squares of first n natural numbers is $ \frac{n(n + 1)(2n + 1)}{6} $; for n = 3, $ 1^2 + 2^2 + 3^2 = \frac{3 \times 4 \times 7}{6} = 14 $.
• The sum of cubes of first n natural numbers is $ \left[\frac{n(n + 1)}{2}\right]^2 $; for n = 4, $ 1^3 + 2^3 + 3^3 + 4^3 = \left[\frac{4 \times 5}{2}\right]^2 = 10^2 = 100 $.
• If a, b, c are in AP, then $ 2b = a + c $; if 2, x, 8 are in AP, then $ 2x = 2 + 8 $ → $ x = 5 $.
• If a, b, c are in GP, then $ b^2 = ac $; if 2, x, 18 are in GP, then $ x^2 = 2 \times 18 = 36 $ → $ x = 6 $.
• The sum $ \sum_{k=1}^{n} k = \frac{n(n+1)}{2} $, $ \sum k^2 = \frac{n(n+1)(2n+1)}{6} $, $ \sum k^3 = \left(\frac{n(n+1)}{2}\right)^2 $ — verify from NCERT.
• In a GP, if $ r = 1 $, then $ S_n = na $; if $ r = -1 $ and n is even, $ S_n = 0 $; if n odd, $ S_n = a $.

Difficulty Level

Intermediate — requires understanding of formulas and their conditional use (e.g., |r| < 1 for infinite sum), but direct application dominates in CUET.

Common CUET Traps

• Trap: Using $ S_\infty = \frac{a}{1 - r} $ for $ |r| \ge 1 $. Avoid: This formula is valid only when $ |r| < 1 $; otherwise, the series diverges.
• Trap: Assuming three terms in AP as a, a + d, a + 2d when symmetry can simplify; Avoid: Use $ a - d, a, a + d $ to reduce variables and simplify equations.
• Trap: Confusing $ a_n $ with $ S_n $ in GP; e.g., using sum formula for term calculation. Avoid: Remember $ a_n = ar^{n-1} $, $ S_n = \frac{a(r^n - 1)}{r - 1} $ — keep them distinct.

Practice MCQs

1. Question: The 10th term of the AP: 5, 9, 13, 17,… is
   A. 37
   B. 41
   C. 45
   D. 49
   Answer: B
   Explanation: $ a = 5, d = 4 $, $ a_{10} = 5 + (10 - 1) \times 4 = 5 + 36 = 41 $.
   Why others fail: Option A (37) comes from using $ n = 9 $ by mistake.

2. Question: The sum of the first 6 terms of a GP with first term 3 and common ratio 2 is
   A. 189
   B. 126
   C. 93
   D. 63
   Answer: A
   Explanation: $ S_6 = \frac{3(2^6 - 1)}{2 - 1} = 3(64 - 1) = 189 $.
   Why others fail: Option B (126) results from using $ r = 3 $ or miscalculating exponent.

3. Question: If the sum to infinity of a GP is 8 and the first term is 4, the common ratio is
   A. $ \frac{1}{4} $
   B. $ \frac{1}{2} $
   C. $ \frac{3}{4} $
   D. $ \frac{2}{3} $
   Answer: B
   Explanation: $ S_\infty = \frac{a}{1 - r} $ → $ 8 = \frac{4}{{1 - r}} $ → $ 1 - r = 0.5 $ → $ r = 0.5 $.
   Why others fail: Option C (3/4) comes from incorrectly solving $ 8 = \frac{4}{1 + r} $.

4. Question: The value of $ 1^2 + 2^2 + 3^2 + \cdots + 10^2 $ is
   A. 385
   B. 425
   C. 465
   D. 505
   Answer: A
   Explanation: $ \sum k^2 = \frac{n(n+1)(2n+1)}{6} = \frac{10 \times 11 \times 21}{6} = 385 $.
   Why others fail: Option B (425) arises from using $ n = 11 $ or arithmetic error.

5. Question: Three numbers in GP have sum 26 and product 216. The largest number is
   A. 6
   B. 8
   C. 12
   D. 18
   Answer: D
   Explanation: Let numbers be $ \frac{a}{r}, a, ar $; product = $ a^3 = 216 $ → $ a = 6 $; sum = $ 6(\frac{1}{r} + 1 + r) = 26 $ → $ \frac{1}{r} + r = \frac{10}{3} $ → solving gives $ r = 3 $ or $ \frac{1}{3} $; largest = $ 6 \times 3 = 18 $.
   Why others fail: Option C (12) comes from assuming AP instead of GP.

Last‑Minute Revision

• ⚠️ $ a_n = a + (n - 1)d $ — AP general term.
• ⚠️ $ S_n = \frac{n}{2}[2a + (n - 1)d] $ — AP sum formula.
• ⚠️ $ a_n = ar^{n-1} $ — GP general term.
• ⚠️ $ S_n = \frac{a(r^n - 1)}{r - 1}, r \ne 1 $ — GP sum.
• ⚠️ $ S_\infty = \frac{a}{1 - r} $ only if $ |r| < 1 $.
• ⚠️ AM between a and b is $ \frac{a + b}{2} $.
• ⚠️ GM between a and b is $ \sqrt{ab} $.
• ⚠️ For three terms in AP: assume $ a - d, a, a + d $.
• ⚠️ For three terms in GP: assume $ \frac{a}{r}, a, ar $.
• ⚠️ $ \sum k = \frac{n(n+1)}{2} $.
• ⚠️ $ \sum k^2 = \frac{n(n+1)(2n+1)}{6} $.
• ⚠️ $ \sum k^3 = \left[\frac{n(n+1)}{2}\right]^2 $.
• ⚠️ If a, b, c in AP → $ 2b = a + c $.
• ⚠️ If a, b, c in GP → $ b^2 = ac $.
• ⚠️ Infinite GP converges only if $ |r| < 1 $.
• ⚠️ First term of GP cannot be zero.
• ⚠️ Common difference d can be negative, zero, or positive.
• ⚠️ Common ratio r can be negative, but $ |r| < 1 $ for convergence.
• ⚠️ Sum of first n odd natural numbers = $ n^2 $.
• ⚠️ $ (1 + 2 + 3 + \cdots + n)^2 = 1^3 + 2^3 + \cdots + n^3 $.