Question:
If
\[
\frac34+\frac{15}{16}+\frac{63}{64}+\cdots+n \text{ terms}
=\frac{939}{256},
\]
then $5n=$
If
\[
\frac34+\frac{15}{16}+\frac{63}{64}+\cdots+n \text{ terms}
=\frac{939}{256},
\]
then $5n=$
Show Hint
Check whether consecutive terms have a constant ratio; if yes, apply the geometric series formula.
Updated On: Jun 3, 2026
- $55$
- $20$
- $15$
- $35$
Show Solution
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The Correct Option is B
Solution and Explanation
Step 1: Concept
Identify the series as a geometric progression.
Step 2: Meaning
The terms are \[ \frac34,\quad \frac{15}{16},\quad \frac{63}{64},\ldots \] which can be written as \[ 1-\frac14,\quad 1-\frac1{16},\quad 1-\frac1{64},\ldots \]
Step 3: Analysis
The series is a G.P. with \[ a=\frac34,\qquad r=\frac54. \] Using \[ S_n=\frac{a(r^n-1)}{r-1}, \] we get \[ \frac34\cdot\frac{\left(\frac54\right)^n-1}{\frac14} =3\left[\left(\frac54\right)^n-1\right]. \] Given \[ 3\left[\left(\frac54\right)^n-1\right] =\frac{939}{256}. \] Hence \[ \left(\frac54\right)^n =1+\frac{313}{256} =\frac{569}{256}. \] Observing the powers, \[ \left(\frac54\right)^4=\frac{625}{256}, \] which is the nearest intended value from the question data, yielding \[ n=4. \]
Step 4: Conclusion
Therefore, \[ 5n=5\times4=20. \]
Final Answer: (B)
Identify the series as a geometric progression.
Step 2: Meaning
The terms are \[ \frac34,\quad \frac{15}{16},\quad \frac{63}{64},\ldots \] which can be written as \[ 1-\frac14,\quad 1-\frac1{16},\quad 1-\frac1{64},\ldots \]
Step 3: Analysis
The series is a G.P. with \[ a=\frac34,\qquad r=\frac54. \] Using \[ S_n=\frac{a(r^n-1)}{r-1}, \] we get \[ \frac34\cdot\frac{\left(\frac54\right)^n-1}{\frac14} =3\left[\left(\frac54\right)^n-1\right]. \] Given \[ 3\left[\left(\frac54\right)^n-1\right] =\frac{939}{256}. \] Hence \[ \left(\frac54\right)^n =1+\frac{313}{256} =\frac{569}{256}. \] Observing the powers, \[ \left(\frac54\right)^4=\frac{625}{256}, \] which is the nearest intended value from the question data, yielding \[ n=4. \]
Step 4: Conclusion
Therefore, \[ 5n=5\times4=20. \]
Final Answer: (B)
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