Visual SolutionPYQ 2024 · Apr 4 Shift 2Standard

Question

If the coefficients of

x^4, x^5

and

x^6

in the expansion of

(1 + x)^n

are in the arithmetic progression, then the maximum value of

n

is:

(A)

7

(B)

21

(C)

28

(D)

14

Solution Path

{}^{n}C_{4}, {}^{n}C_{5}, {}^{n}C_{6}

in AP gives

n^2 - 21n + 98 = 0

, so

n = 7

14

. Max

n = 14

01Question Setup

1/4

Coefficients of

x^4, x^5, x^6

(1+x)^n

are in arithmetic progression. Find max

n

{}^{n}C_{4},\; {}^{n}C_{5},\; {}^{n}C_{6}

in AP

02AP Condition

2/4

For three terms in AP:

2 \cdot {}^{n}C_{5} = {}^{n}C_{4} + {}^{n}C_{6}

. Use ratio formula

\frac{{}^{n}C_{r}}{{}^{n}C_{r-1}} = \frac{n-r+1}{r}

2 \cdot {}^{n}C_{5} = {}^{n}C_{4} + {}^{n}C_{6}

03Solve QuadraticKEY INSIGHT

3/4

Simplify to

n^2 - 21n + 98 = 0

. Factor:

(n-14)(n-7) = 0

, giving

n = 7

n = 14

(n-14)(n-7) = 0

04Final Answer

4/4

Maximum value of

n

14

\boxed{14}

Concepts from this question1 concepts unlocked

★

EASY

T(r+1) = C(n,r) * a^(n-r) * b^r gives any specific term without expanding fully

T_{r+1} = {}^nC_r\, a^{n-r}\, b^r

Core formula for finding specific terms, term independent of x, rational terms, etc. Used in 4-5 questions every JEE paper.

Specific termIndependent termRational/integral termsCoefficient finding

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