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Sequences and Series question

2002 · Shift 0 · Q112
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Sequences and Series question

2002 · Shift 0 · Q112

JEE MainMathematicsSequences and SeriesMCQ+4 / −1
Fifth term of a GP is 2, then the product of its 9 terms is
  1. A
    256
  2. B
    512
  3. C
    1024
  4. D
    none of these
View written solutionFree

Correct answer: B

  1. Let the 9 terms of the GP be a,ar,ar2,ar3,ar4,ar5,ar6,ar7,ar8a, ar, ar^2, ar^3, ar^4, ar^5, ar^6, ar^7, ar^8a,ar,ar2,ar3,ar4,ar5,ar6,ar7,ar8

  2. The 5th term is given as 2. Since the 5th term is ar4=2ar^4 = 2ar4=2

  3. We need the product of all 9 terms: P=a⋅ar⋅ar2⋅ar3⋅ar4⋅ar5⋅ar6⋅ar7⋅ar8P = a \cdot ar \cdot ar^2 \cdot ar^3 \cdot ar^4 \cdot ar^5 \cdot ar^6 \cdot ar^7 \cdot ar^8P=a⋅ar⋅ar2⋅ar3⋅ar4⋅ar5⋅ar6⋅ar7⋅ar8

  4. Add the powers of rrr: P=a9r0+1+2+3+4+5+6+7+8=a9r36P = a^9 r^{0+1+2+3+4+5+6+7+8} = a^9 r^{36}P=a9r0+1+2+3+4+5+6+7+8=a9r36

  5. Rewrite this as: P=(ar4)9P = (ar^4)^9P=(ar4)9 because a9r36=(ar4)9a^9 r^{36} = (ar^4)^9a9r36=(ar4)9

  6. Substitute ar4=2ar^4 = 2ar4=2: P=29=512P = 2^9 = 512P=29=512

  7. Therefore, the product of the 9 terms is 512\boxed{512}512​

  8. Option check:

    • A: 256256256 ❌
    • B: 512512512 ✅
    • C: 102410241024 ❌
    • D: none of these ❌

Hence, the correct option is B.

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