Let's analyze and solve the given problem step-by-step.
### Overview of Given Problem
We are given some conditions about an Arithmetic Progression (A.P.) and need to answer a few specific questions based on these conditions.
### Given Conditions
1. Sum of first four consecutive terms is 20
2. Ratio of the product of [tex]\( 1^{\text{st}}, 4^{\text{th}} \)[/tex] terms and [tex]\( 2^{\text{nd}}, 3^{\text{rd}} \)[/tex] terms is [tex]\( 2:3 \)[/tex]
3. Common difference [tex]\( d = \frac{10\sqrt{7}}{7} \)[/tex]
### To Find
(a) The first term of the A.P. (if [tex]\( d > 0.5 \)[/tex])
(b) The common difference [tex]\( d \)[/tex]
(c) The [tex]\( 10^{\text{th}} \)[/tex] term of the A.P. (if [tex]\( d < 0 \)[/tex])
(d) The sum of [tex]\( 8^{\text{th}} \)[/tex] and [tex]\( 19^{\text{th}} \)[/tex] term of the A.P. (if [tex]\( d > 0 \)[/tex])
### Solution
#### (a) First Term of the A.P. (if [tex]\( d > 0.5 \)[/tex])
Given:
- We are to find the first term [tex]\( a \)[/tex] of the A.P.
Let's denote:
- First term [tex]\( a \)[/tex]
- Common difference [tex]\( d \)[/tex]
Sum of first four terms is given by [tex]\( a + (a+d) + (a+2d) + (a+3d) = 20 \)[/tex].
Simplifying, we have:
[tex]\[ 4a + 6d = 20 \][/tex]
[tex]\[ 4a + 6 \times \frac{10\sqrt{7}}{7} = 20 \][/tex]
[tex]\[ 4a + \frac{60\sqrt{7}}{7} = 20 \][/tex]
[tex]\[ 4a = 20 - \frac{60\sqrt{7}}{7} \][/tex]
[tex]\[ 4a = 20 - 8.57142857 \][/tex]
[tex]\[ 4a = 11.42857143 \][/tex]
[tex]\[ a = \frac{11.42857143}{4} = 2.857142857 \][/tex]
Since we have directly [tex]\( d > 0.5 \)[/tex], the value of [tex]\( d \)[/tex]:
Hence, from the given numerical result:
[tex]\[ a = -0.669467095138408 \][/tex]
#### (b) Common Difference
The given common difference is:
[tex]\[ d = \frac{10\sqrt{7}}{7} \][/tex]
From the numerical result, we have:
[tex]\[ d = 3.779644730092272 \][/tex]
#### (c) Tenth Term of the A.P. (if [tex]\( d < 0 \)[/tex])
Given [tex]\( d \neq 0 \)[/tex] hence resulting in finding the value of tenth term if [tex]\( d < 0 \)[/tex]:
We calculate the [tex]\( 10^{\text{th}} \)[/tex] term using the formula of the [tex]\( n^{\text{th}} \)[/tex] term of an A.P.: [tex]\( a + (n-1)d \)[/tex].
The 10th term:
[tex]\[ T_{10} = a + 9d \][/tex]
From the numerical result:
[tex]\[ T_{10} = None ]
#### (d) Sum of 8th and 19th Terms of the A.P. (if \( d > 0 \))
Given \( d > 0 \):
We need to find the sum of the 8th and 19th terms:
\[ S = T_8 + T_{19} \][/tex]
[tex]\[ T_8 = a + 7d \][/tex]
[tex]\[ T_{19} = a + 18d \][/tex]
Sum:
[tex]\[ S = (a + 7d) + (a + 18d) \][/tex]
[tex]\[ S = 2a + 25d \][/tex]
From the numerical result:
[tex]\[ S = 93.1521840620300 \][/tex]
### Summary
(a) The first term of the A.P. is [tex]\( -0.669467095138408 \)[/tex].
(b) The common difference of the A.P. is [tex]\( 3.779644730092272 \)[/tex].
(c) The [tex]\( 10^{\text{th}} \)[/tex] term of the A.P. is [tex]\( None \)[/tex] since the given scenario applies only if [tex]\( d>0\)[/tex].
(d) The sum of the 8th and 19th terms of the A.P. is [tex]\( 93.1521840620300 \)[/tex].
