Given that the 7th term of AP is 3 greater than the 5th term,the sum of the 10th term and 3th term is 45, calculate,20th term,8 term and the sum of the first five terms?.
so, use what they told you:
a+6d = a+4d + 3
a+9d + a+2d = 45
3th term? Is that a typo for 3rd, or for 13th? I assumed 3rd.
Solve for a and d, and then just find
a+19d, a+7d, and S5 = 5/2 (2a+4d)
To solve this problem, we need to use the properties of an arithmetic progression (AP).
An AP is a sequence of numbers in which the difference between any two consecutive terms is constant.
Let's proceed step by step:
Step 1: Find the common difference (d)
Given that the 7th term is 3 greater than the 5th term, we can express this as:
a + 6d = a + 4d + 3, where 'a' is the first term and 'd' is the common difference.
By simplifying the equation, we get: 6d - 4d = 3
Simplifying further, we have: 2d = 3
So, the common difference (d) is 3/2 or 1.5.
Step 2: Find the first term (a)
To find the first term (a), we need to use the equation a + 6d = 7th term.
Given that the 7th term is a + 6d, we can express this as:
a + 6(1.5) = a + 4(1.5) + 3
Simplifying the equation, we get: a + 9 = a + 6 + 3
Simplifying further, we have: a + 9 = a + 9
This equation is true, indicating that the value of 'a' can be anything.
Step 3: Find the 20th term
Using the formula for the nth term of an AP, we can find the 20th term:
nth term = a + (n - 1)d
Plugging in the values, we have: 20th term = a + (20 - 1) * 1.5
Simplifying the equation, we get: 20th term = a + 19 * 1.5
Simplifying further, we have: 20th term = a + 28.5
Step 4: Find the 8th term
Using the same formula, we can find the 8th term:
8th term = a + (8 - 1) * 1.5
Simplifying this equation gives us: 8th term = a + 10.5
Step 5: Find the sum of the first five terms
The sum of the first 'n' terms of an AP is given by the formula:
Sum = (n/2)[2a + (n - 1)d]
Plugging in the values, we can find the sum of the first five terms:
Sum = (5/2)[2a + (5 - 1) * 1.5]
Simplifying the equation gives us: Sum = (5/2)[2a + 6]
Further simplification leads to: Sum = (5/2)(2a + 6)
Finally, we have: Sum = 5a + 15
Using these steps, we can calculate the 20th term, 8th term, and the sum of the first five terms. However, we cannot determine the value of these terms without knowing the value of 'a' (the first term).