Quadratic Equations and Sequence Series Questions for AGNIVEER NAVY

Quadratic Equations — Standard Form and Roots

A quadratic equation is any equation of the form ax^2 + bx + c = 0 where a is not zero. The solutions (roots) tell you where the parabola crosses the x-axis. In AGNIVEER_NAVY, you'll rarely need to graph; instead, you'll find roots using the quadratic formula or factorization, then use those roots to answer questions about the equation itself. The discriminant (b^2 - 4ac) tells you how many real roots exist without solving.

• Standard form: ax^2 + bx + c = 0 (a ≠ 0)
• Discriminant D = b^2 - 4ac determines nature of roots: D > 0 (two distinct real), D = 0 (one repeated real), D < 0 (no real roots)
• Roots are real and rational if D is a perfect square
• If roots are α and β, then α + β = -b/a and αβ = c/a (Vieta's formulas — most useful shortcut)
• Factorization method works when roots are integers; quadratic formula works always

Arithmetic Progression (AP)

An arithmetic progression is a sequence where consecutive terms have a constant difference (called common difference d). Examples: 2, 5, 8, 11 (d=3) or 10, 7, 4, 1 (d=-3). AGNIVEER_NAVY tests your ability to find the nth term, sum of n terms, and solve word problems (like finding how many terms, or which term equals a given value). AP is the most frequently tested sequence type.

• First term = a, common difference = d
• nth term: T_n = a + (n-1)d
• Sum of first n terms: S_n = n/2 × (2a + (n-1)d) or S_n = n/2 × (first term + last term)
• If you know any three of {a, d, n, T_n, S_n}, you can find the rest
• Identify AP by checking if differences between consecutive terms are equal

Geometric Progression (GP)

A geometric progression is a sequence where consecutive terms have a constant ratio (called common ratio r). Examples: 2, 6, 18, 54 (r=3) or 100, 50, 25, 12.5 (r=0.5). AGNIVEER_NAVY asks fewer GP questions than AP, but when they appear, they test nth term, sum of n terms, and infinite series (if |r| < 1). GP grows or shrinks exponentially, so watch for large numbers or fractions.

• First term = a, common ratio = r
• nth term: T_n = a × r^(n-1)
• Sum of first n terms: S_n = a × (r^n - 1) / (r - 1) when r ≠ 1
• Sum of infinite GP (if |r| < 1): S_∞ = a / (1 - r)
• If r > 1, GP increases; if 0 < r < 1, GP decreases; if r < 0, terms alternate in sign

Connecting Quadratics to Sequences

AGNIVEER_NAVY sometimes asks questions where a quadratic equation arises from a sequence problem. For example, finding which term of an AP equals a given value leads to solving a quadratic. Or, if the sum of n terms of an AP equals some value, you set up and solve a quadratic in n. This is a high-value skill because it combines two topics and tests problem-solving depth.

• If T_n = k (a specific value), substitute T_n = a + (n-1)d = k and solve for n (may be quadratic)
• If S_n = k, substitute S_n = n/2 × (2a + (n-1)d) = k, simplify, and solve (often quadratic in n)
• Always reject negative or non-integer values of n (n must be a positive integer)
• Check your answer by substituting back into the original sequence formula

• Forgetting that n must be a positive integer. If solving a quadratic gives n = -3 or n = 2.5, reject it immediately.
• Confusing the sum formula for AP: using S_n = n/2 × (2a + (n-1)d) but making arithmetic errors. Double-check by using the alternative S_n = n/2 × (first + last).
• In GP, using the sum formula S_n = a × (r^n - 1) / (r - 1) when r = 1. If r = 1, all terms are equal, so S_n = n × a.
• Misidentifying the common difference or ratio. Always verify by computing at least two consecutive differences or ratios before proceeding.
• In quadratics, applying Vieta's formulas incorrectly. Remember: sum = -b/a (note the negative sign), product = c/a. Sign errors are frequent.

• AP = Arithmetic (addition/subtraction), GP = Geometric (multiplication/division). Think: AP is linear growth, GP is exponential growth.
• For Vieta: Sum = -b/a (negative b), Product = c/a (positive c). Mnemonic: 'Sum is Negative, Product is Positive'.
• T_n formulas: AP uses (n-1)d (linear), GP uses r^(n-1) (exponential). The power grows with n in GP, not in AP.
• Discriminant D: D > 0 (two roots), D = 0 (one root), D < 0 (no real roots). Think: D is the 'Decider' of root count.

• AGNIVEER_NAVY papers typically include 1-2 AP questions (finding nth term or sum) and 0-1 GP question. AP is higher priority.
• Quadratic questions often appear as 'find the roots' or 'if roots are α and β, find α + β or αβ'. Use Vieta's formulas to save time — don't solve the full quadratic unless asked.
• Word problems combining quadratics and sequences are common. Example: 'A sum increases by a fixed amount each month. After how many months will it reach Rs. 5000?' Set up the AP formula, solve the resulting quadratic.
• Time management: AP/GP questions should take 1-2 minutes each. If you're stuck, move on and return later. These are not trick questions — if your setup is correct, arithmetic is straightforward.
• Recent papers show a trend toward 'find the common difference' or 'find the first term' given partial information. Practice reverse-engineering sequences from limited data.