Let a_1$a_{1}$ , a_2$a_{2}$ , a_3$a_{3}$ , ..... be a G.P. of increasing positive terms such that a_2 cdot a_3 cdot a_4 = 64$a_{2} \cdot a_{3} \cdot a_{4} = 64$ and a_1 + a_3 + a_5 = frac8137$a_{1} + a_{3} + a_{5} = \frac{813}{7}$ . Then a_3 + a_5 + a_7$a_{3} + a_{5} + a_{7}$ is equal to :

### Related Formula ntext-th term of a GP: T_n = a r^n-1$$n\text{-th term of a GP}: T_n = a r^{n-1}$$ ### Core Logic Let the terms of the GP be a, ar, ar^2, ar^3, dots$a, ar, ar^2, ar^3, \dots$ Given: a_2 cdot a_3 cdot a_4 = 64$a_2 \cdot a_3 \cdot a_4 = 64$ (ar) cdot (ar^2) cdot (ar^3) = 64$$(ar) \cdot (ar^2) \cdot (ar^3) = 64$$ a^3 r^6 = 64 Rightarrow (ar^2)^3 = 64 Rightarrow ar^2 = 4$$a^3 r^6 = 64 \Rightarrow (ar^2)^3 = 64 \Rightarrow ar^2 = 4$$ Since it is a GP of increasing positive terms, r > 1$r > 1$ and a > 0$a > 0$. ### Step 1: Utilize the sum condition Given: a_1 + a_3 + a_5 = frac8137$a_1 + a_3 + a_5 = \frac{813}{7}$ a + ar^2 + ar^4 = frac8137$$a + ar^2 + ar^4 = \frac{813}{7}$$ Extract a$a$ from ar^2 = 4 Rightarrow a = frac4r^2$ar^2 = 4 \Rightarrow a = \frac{4}{r^2}$. Substitute this in: frac4r^2 + 4 + 4r^2 = frac8137$$\frac{4}{r^2} + 4 + 4r^2 = \frac{813}{7}$$ 4left(frac1r^2 + 1 + r^2right) = frac8137$$4\left(\frac{1}{r^2} + 1 + r^2\right) = \frac{813}{7}$$ Let r^2 = t$r^2 = t$: 4left(frac1t + 1 + tright) = frac8137$$4\left(\frac{1}{t} + 1 + t\right) = \frac{813}{7}$$ Wait, there is a much faster method by just scaling the required expression. ### Step 2: Calculate the required expression We need a_3 + a_5 + a_7 = ar^2 + ar^4 + ar^6$a_3 + a_5 + a_7 = ar^2 + ar^4 + ar^6$. Notice that ar^2 + ar^4 + ar^6 = r^2 (a + ar^2 + ar^4)$ar^2 + ar^4 + ar^6 = r^2 (a + ar^2 + ar^4)$. So, required sum = r^2 left(frac8137right)$r^2 \left(\frac{813}{7}\right)$. To find r^2$r^2$, we solve the quadratic in t = r^2$t = r^2$: 4left(fract^2 + t + 1tright) = frac8137$$4\left(\frac{t^2 + t + 1}{t}\right) = \frac{813}{7}$$ 28t^2 + 28t + 28 = 813t$$28t^2 + 28t + 28 = 813t$$ 28t^2 - 785t + 28 = 0$$28t^2 - 785t + 28 = 0$$ The roots are t = 28$t = 28$ and t = frac128$t = \frac{1}{28}$. Since the GP is increasing, r > 1 Rightarrow r^2 = 28$r > 1 \Rightarrow r^2 = 28$. ### Step 3: Final evaluation Alternatively, expand directly: ar^2 (1 + r^2 + r^4) = 4 (1 + 28 + (28)^2) = 4(1 + 28 + 784) = 4(813) = 3252.$$ar^2 (1 + r^2 + r^4) = 4 (1 + 28 + (28)^2) = 4(1 + 28 + 784) = 4(813) = 3252.$$ (Note: r^2 times frac8137 = 28 times frac8137 = 4 times 813 = 3252$r^2 \times \frac{813}{7} = 28 \times \frac{813}{7} = 4 \times 813 = 3252$) ### Pattern Recognition In GP questions demanding a sum shifted by a fixed index (like a_1+a_3+a_5$a_1+a_3+a_5$ to a_3+a_5+a_7$a_3+a_5+a_7$), immediately look to factor out the common ratio multiplier r^k$r^k$. Here it's a simple scaling by r^2$r^2$. ### Evaluation Rubric / Model Answer null ### Chapter Mix Class 11 Maths: Sequences and Series Class 10 Maths: Quadratic Equations