Sequences & Series
Geometric progression terms
nta_pyq_2025_apr
Grade 11
Question:
Let $a_1, a_2, a_3, \ldots$ be a G. P. of increasing positive numbers. If $a_2 a_3 = 729$ and $a_3 + a_5 = \frac{111}{4}$, then $24(a_1+a_2+a_3)$ is equal to
Step-by-Step Solution
Key Concept: For the increasing positive GP, use$a_n=ar^{n-1}$and solve$product/sum$equations for$a,r$.
Let the I st term of G.P. be a \& common ratio be r$(3)$2 4$a_{3}$$a_{5} = ar$$\cdot$ar = 729$2$6 = a$$r = 729$$3 = ar = 27$.... (i) 111 3$$a_{2}$+$a_{4}$= ar + ar = 4$$3 = ar =.$... (ii) 4 (i)$$\div$ (ii) 3 ar$27 = ar$$3/4$2$r = 36$$r = 6$from (ii) 3 1 a$(6)$= $\Rightarrow$$a = 4$8 Now, 24 ($a + a + a$) 1 2 3$2 = 24$($a + ar + a$)$2 = 24a$($1 + r + r$)$1 = 24$$\times$ ($1 + 6 + 36)$$8 = 3$(43)$= 129$
<div class="key-concept"><strong>Key Concept:</strong> For the increasing positive GP, use$a_n=ar^{n-1}$and solve$product/sum$equations for$a,r$.</div>
<div class="trap-box"><strong>Trap:</strong> Do not take the negative value of the common ratio because the terms are increasing and positive.</div>
Correct Answer: 3