The sum of 11 of an a.p is 891 find the 28 and 45 if the common difference is 15
Poor wording.
The sum of 11 what? 11 terms? I will assume that
You must know your 2 main formulas for an AP
term(n terms) and sum(n terms)
sum(11) = 11/2(2a + 10d), but you told me that sum(11) = 891 and d = 15
so let's sub that into the above
891 = (11/2)(2a + 150)
891 = 11(a + 75)
891 = 11a + 825
take over to find a, then use your definition of term(28) and term(45) to finish up
To find the 28th and 45th terms of an arithmetic progression (AP), we need to use the formula for the nth term of an arithmetic progression:
𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑
where 𝑎𝑛 is the nth term, 𝑎1 is the first term, 𝑛 is the position of the term, and 𝑑 is the common difference.
Given that the sum of the first 11 terms is 891 and the common difference is 15, we can find the first term, 𝑎1, and calculate the required terms.
Step 1: Calculate the first term (𝑎1) using the sum of the first 11 terms (𝑆11).
The sum of the first n terms of an arithmetic progression is given by the formula:
𝑆𝑛 = (𝑛/2)(2𝑎1 + (𝑛 − 1)𝑑)
Substituting the values given: 𝑆11 = 891 and 𝑑 = 15, we can solve for 𝑎1.
891 = (11/2)(2𝑎1 + 10 × 15)
891 = 55𝑎1 + 1500
55𝑎1 = 891 − 1500
55𝑎1 = -609
𝑎1 = -609/55
𝑎1 ≈ -11.07
Therefore, the first term, 𝑎1, is approximately -11.07.
Step 2: Calculate the 28th term (𝑎28).
Using the formula 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑, substitute 𝑎1 = -11.07, 𝑑 = 15, and 𝑛 = 28 to find 𝑎28.
𝑎28 = -11.07 + (28 − 1) × 15
𝑎28 = -11.07 + 27 × 15
𝑎28 = -11.07 + 405
𝑎28 ≈ 393.93
Therefore, the 28th term, 𝑎28, is approximately 393.93.
Step 3: Calculate the 45th term (𝑎45).
Using the same formula 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑, substitute 𝑎1 = -11.07, 𝑑 = 15, and 𝑛 = 45 to find 𝑎45.
𝑎45 = -11.07 + (45 − 1) × 15
𝑎45 = -11.07 + 44 × 15
𝑎45 = -11.07 + 660
𝑎45 ≈ 648.93
Therefore, the 45th term, 𝑎45, is approximately 648.93.
In conclusion, the 28th term of the arithmetic progression is approximately 393.93, and the 45th term is approximately 648.93.
To find the 28th and 45th term of an arithmetic progression (AP) given the common difference, we can use the formula for the nth term of an AP:
nth term = a + (n - 1)d
Where:
- nth term is the term you want to find
- a is the first term of the AP
- n is the position of the term in the AP
- d is the common difference of the AP
In this case, we are given the sum of 11 terms of the AP as 891 and the common difference as 15.
To find the value of a, we can use the formula for the sum of an AP:
Sum = (n/2)(2a + (n - 1)d)
Plugging in the given values, we have:
891 = (11/2)(2a + (11 - 1)(15))
Now we can solve this equation to find the value of a.
891 = (11/2)(2a + 10(15))
891 = (11/2)(2a + 150)
891 = (11/2)(2a + 150)
891 = 11a + 825
891 - 825 = 11a
66 = 11a
a = 6
Now that we have the value of a, we can find the 28th term:
28th term = 6 + (28 - 1)(15)
28th term = 6 + 27 * 15
28th term = 6 + 405
28th term = 411
Similarly, we can find the 45th term:
45th term = 6 + (45 - 1)(15)
45th term = 6 + 44 * 15
45th term = 6 + 660
45th term = 666
Therefore, the 28th term is 411, and the 45th term is 666.