Question

Q2 In a finite GP of 7 positive terms, if the sum of the first 3 terms is 7 and the sum of the last 3 terms is 112, then the middle term of the GP is equal to​

Answer 1

Step-by-step explanation:

Given:In a finite GP of 7 positive terms, if the sum of the first 3 terms is 7 and the sum of the last 3 terms is 112.

To find: The middle term of the GP is equal to ?

Solution:

Let the first term of GP is 'a' and

common ratio is 'r'.

Step 1: Form equations by putting the given conditions

ATQ

Sum of first three terms are 7. So,

$a + ar + a {r}^{2} = 7 \\ \\ or \\ \\ a(1 + r + {r}^{2} ) = 7 \: \: \: ...eq1$

and

Sum of last three terms are 112. So,

$a {r}^{4} + a {r}^{5} + a {r}^{6} = 112 \\ \\ or \\ \\ a {r}^{4} (1 + r + {r}^{2} ) = 112 \: \: \: ...eq2$

Step 2: Put the value from eq1 into eq2 to find value of 'r'

${r}^{4} a(1 + r + {r}^{2} ) = 112 \\ \\ {r}^{4} \times 7 = 112 \\ \\{r}^{4} = \frac{112}{7} \\ \\ {r}^{4} = 16 \\ \\ {r}^{4} = {2}^{4}$

powers are same,thus compare base

$r = 2$

Step 3: Put the value of r in eq1 and find value of 'a'

$a(1 + 2 + {2}^{2} ) = 7 \\ \\ a(1 + 2 + 4) = 7 \\ \\ 7a = 7 \\ \\ a = \frac{7}{7} \\ \\ a = 1$

Step 4: Find the middle term

In seven terms fourth term is middle term.

$a {r}^{3} = 1 \times {2}^{3} \\ \\ a {r}^{3}= 8$

Final answer:

Middle term of GP is 8.

Hope it helps you.

To learn more:

find five numbers in GP such that their sum is 31/4 and product is 1

https://brainly.in/question/18181027