the sum of 8th terms of an AP is 549 while the common difference is 3. find the 56th term and the sum of it's 32 terms
just plug in your formulas. If you mean
"the sum of the first 8 terms" then
8/2 (2a + 7d) = 549
You have d, so find a, and then
a + 55d
and
32/2 (2a + 31d)
To find the 8th term of an arithmetic progression (AP), we will use the formula:
aₙ = a₁ + (n - 1)d
where:
aₙ is the nth term,
a₁ is the first term,
n is the number of terms, and
d is the common difference.
Given that the sum of the 8th terms of the AP is 549 and the common difference is 3, we can substitute these values into the formula and solve for a₁:
549 = a₁ + (8 - 1) * 3
549 = a₁ + 7 * 3
549 = a₁ + 21
a₁ = 549 - 21
a₁ = 528
So the first term (a₁) is 528.
To find the 56th term, we use the same formula but substitute n = 56 and a₁ = 528:
a₅₆ = 528 + (56 - 1) * 3
a₅₆ = 528 + 55 * 3
a₅₆ = 528 + 165
a₅₆ = 693
Therefore, the 56th term of the AP is 693.
To find the sum of the first 32 terms, we can use the formula for the sum of an arithmetic series:
Sₙ = (n/2)(2a₁ + (n - 1)d)
where:
Sₙ is the sum of the first n terms.
Plugging in the known values, we have:
S₃₂ = (32/2)(2 * 528 + (32 - 1) * 3)
S₃₂ = 16(1056 + 31 * 3)
S₃₂ = 16(1056 + 93)
S₃₂ = 16(1149)
S₃₂ = 18384
Therefore, the sum of the first 32 terms of the AP is 18384.
To find the sum of the 8th terms of an arithmetic progression (AP), we need to know the formula to calculate the sum of an AP. The formula is:
Sum of AP = (n/2) * (2a + (n-1)d)
Where:
- n is the number of terms
- a is the first term
- d is the common difference
In this case, we are given:
- The sum of the 8th term is 549
- The common difference is 3
So, let's plug in the values into the formula and solve for the sum of the 8th terms:
549 = (8/2) * (2a + (8-1) * 3)
Simplifying,
549 = 4 * (2a + 7 * 3)
549 = 4 * (2a + 21)
549 = 8a + 84
8a = 549 - 84
8a = 465
a = 465/8
a = 58.125
Therefore, the value of the first term (a) is 58.125.
We can find the 56th term of the AP using the formula:
nth term = a + (n-1) * d
Substituting a = 58.125, d = 3, and n = 56:
56th term = 58.125 + (56-1) * 3
56th term = 58.125 + 55 * 3
56th term = 58.125 + 165
56th term = 223.125
Therefore, the value of the 56th term is 223.125.
To find the sum of the first 32 terms, we can use the formula for the sum of an AP:
Sum of AP = (n/2) * (2a + (n-1)d)
Substituting a = 58.125, d = 3, and n = 32:
Sum of 32 terms = (32/2) * (2 * 58.125 + (32-1) * 3)
Sum of 32 terms = 16 * (116.25 + 31 * 3)
Sum of 32 terms = 16 * (116.25 + 93)
Sum of 32 terms = 16 * 209.25
Sum of 32 terms = 3348
Therefore, the sum of the first 32 terms is 3348.