Question

Let < a>be an arithmetic sequence whose first term is 1 and < bn> be a geometric sequence whose first term is 2. If the common ratio of geometric sequence is half the common different of arithmetic sequence, then the minimum value of (a4b1 + a3b2 + 2a1 b3) is equal to

Answer 1

Given :

Let < a>be an arithmetic sequence whose first term is 1 and < bn> be a geometric sequence whose first term is 2. If the common ratio of geometric sequence is half the common different of arithmetic sequence.

To Find:

minimum value of (a4b1 + a3b2 + 2a1 b3)

Step-by-step explanation:

• <a> is An arithmetic progression where $a_{1}=1$ and common difference be $d$ then the terms of the Arithmetic Progression .$a_{1}=1\\a_{2}=1+d\\a_{3}=1+2\times d\\a_{4}=1+3\times d$

• <b> is an Geometric Progression where common ratio is $r=d\div 2$ $b_{1}=2\\b_{2}=2\times (d\div 2)\\ b_{3}=2\times (d\div 2)^{2} \\b_{4}=2\times (d\div 2)^{3}$
• The minimum value of $\\(a_{4}\times b_{1} + a_{3}\times b_{2} + 2\times a_{1}\times b_{3})\\=(1+3\times d)\times 2 + (1+2\times d)\times (2\times (d\div 2)) + 2\times 1\times (2\times (d\div 2)^{2} )\\=2+6\times d +d+ 2\times d^{2} +d^{2}\\ =3\times d^{2} +7\times d +2$
• For finding the minimum value of expression let us find the derivative of expression $z=3\times d^{2}+ 7\times d+2\\\\ \[ \frac{\partial z}{\partial d}=6\times d+7=0\\\\d=-7\div 6$
• Now substitute the value of d in expression we get $z=3\times d^{2} +7\times d+2\\z=3\times(49\div 36) -49\div 6 +2\\z=-25\div 12$

The final answer is $-25\div 12$