Sequences & Series
Sum of G.P.
Grade 11
Question:
<p>Let \(A_r\), \(r = 1, 2, 3, \ldots\) be points on the number line such that \(OA_1, OA_2, OA_3, \ldots\) are in G.P., where O is origin and the common ratio of the G.P. be a positive proper fraction. Let \(M_r\) be the middle point of the line segment \(A_r A_{r+1}\). Then the value \(\sum_{r=1}^{\infty} OM_r\) is equal to</p>
<p>(a) \(\dfrac{OA_1(OSA_1 - OA_2)}{2(OA_1 + OA_2)}\)</p>
<p>(b) \(\dfrac{OA_1(OA_1 - OA_2)}{2(OA_1 + OA_2)}\)</p>
<p>(c) \(\dfrac{OA_1}{2(OA_1 - OA_2)}\)</p>
<p>(d) \(\infty\)</p>
Step-by-Step Solution
Key Concept: Express the position of midpoint $M_r$ in terms of the G.P. terms, then sum the infinite series using the formula for sum of an infinite G.P. The midpoint $M_r$ lies at $\frac{OA_r + OA_{r+1}}{2}$, where distances form a G.P. with first term $a$ and common ratio $q$ (where $0 < q < 1$).
<p><strong>Step 1: Set up the G.P.</strong></p><p>Let $OA_r = aq^{r-1}$ where $a = OA_1$ and $0 < q < 1$ is the common ratio.</p><p><strong>Step 2: Find $OM_r$.</strong></p><p>$M_r$ is the midpoint of segment $A_r A_{r+1}$, so:</p><p>$$OM_r = \frac{OA_r + OA_{r+1}}{2} = \frac{aq^{r-1} + aq^r}{2} = \frac{aq^{r-1}(1+q)}{2}$$</p><p><strong>Step 3: Sum the infinite series.</strong></p><p>$$\sum_{r=1}^{\infty} OM_r = \sum_{r=1}^{\infty} \frac{aq^{r-1}(1+q)}{2} = \frac{a(1+q)}{2} \sum_{r=1}^{\infty} q^{r-1}$$</p><p><strong>Step 4: Apply G.P. sum formula.</strong></p><p>$$\sum_{r=1}^{\infty} q^{r-1} = \frac{1}{1-q}$$ (since $|q| < 1$)</p><p><strong>Step 5: Final answer.</strong></p><p>$$\sum_{r=1}^{\infty} OM_r = \frac{a(1+q)}{2} \cdot \frac{1}{1-q} = \frac{a(1+q)}{2(1-q)}$$</p><p>∴ Answer: C</p>
Correct Answer: C