Sure, let's go step-by-step to solve this question.
### Given Data:
- Savings in the 1st month: Rs. 32000
- Savings in the 2nd month: Rs. 36000
- Savings in the 3rd month: Rs. 40000
The savings form an arithmetic sequence where:
- [tex]\( a_1 = 32000 \)[/tex]
- [tex]\( a_2 = 36000 \)[/tex]
- [tex]\( a_3 = 40000 \)[/tex]
### a. Find the common difference of the sequence:
The common difference [tex]\( d \)[/tex] in an arithmetic sequence is given by:
[tex]\( d = a_2 - a_1 \)[/tex]
So,
[tex]\( d = 36000 - 32000 = 4000 \)[/tex]
Answer: [tex]\( d = 4000 \)[/tex]
### b. Find the saving in the 4th month:
The savings in the 4th month [tex]\( a_4 \)[/tex] can be found using the formula of the arithmetic sequence:
[tex]\( a_4 = a_3 + d \)[/tex]
So,
[tex]\( a_4 = 40000 + 4000 = 44000 \)[/tex]
Answer: [tex]\( a_4 = 44000 \)[/tex]
### c. Find the total saving in 3 years:
The total number of months in 3 years is:
[tex]\( 3 \times 12 = 36 \)[/tex] months
The sum [tex]\( S_n \)[/tex] of the first [tex]\( n \)[/tex] terms of an arithmetic sequence is given by:
[tex]\[ S_n = \frac{n}{2} [2a + (n-1)d] \][/tex]
For [tex]\( n = 36 \)[/tex],
[tex]\[ S_{36} = \frac{36}{2} [2 \times 32000 + (36-1) \times 4000] \][/tex]
[tex]\[ S_{36} = 18 [64000 + 140000] \][/tex]
[tex]\[ S_{36} = 18 \times 204000 \][/tex]
So,
[tex]\[ S_{36} = 3672000 \][/tex]
Answer: Rs. 3672000
### d. In how many months will she save Rs. 2000000?
We need to find the number of months [tex]\( n \)[/tex] it will take to save Rs. 2000000.
Using the sum formula for an arithmetic sequence:
[tex]\[ \frac{n}{2} [2a + (n-1)d] = 2000000 \][/tex]
[tex]\[ \frac{n}{2} [2 \times 32000 + (n-1) \times 4000] = 2000000 \][/tex]
[tex]\[ n(64000 + 4000(n-1)) = 4000000 \][/tex]
[tex]\[ n(64000 + 4000n - 4000) = 4000000 \][/tex]
[tex]\[ n(60000 + 4000n) = 4000000 \][/tex]
This simplifies to:
[tex]\[ 4000n^2 + 60000n - 4000000 = 0 \][/tex]
[tex]\[ n^2 + 15n - 1000 = 0 \][/tex]
Solving the quadratic equation, we get [tex]\( n = 25 \)[/tex], as it is the positive integer solution.
Answer: 25 months
