Given that the first term of an Ap is 7and its 10th term is twice the 2nd term, calculate the( A 19th term (b sum of 28th
To find the 19th term of the arithmetic progression (AP), we need to determine the common difference (d) between consecutive terms. Given that the first term (a₁) is 7 and the 10th term (a₁₀) is twice the second term (a₂), we can use this information to find the common difference.
Step 1: Find the common difference (d)
a₁₀ = a₁ + (10 - 1) * d
2a₂ = a₁ + (2 - 1) * d
Substituting the values:
7 + 9d = 2(7 + d)
Distribute 2 to the terms in the parentheses:
7 + 9d = 14 + 2d
Subtract 7 from both sides:
9d = 7 + 2d - 7
9d - 2d = 7
7d = 7
Divide both sides by 7:
d = 1
Therefore, the common difference (d) is 1.
Step 2: Find the 19th term (a₁₉)
a₁₉ = a₁ + (19 - 1) * d
a₁₉ = 7 + 18 * 1
a₁₉ = 7 + 18
a₁₉ = 25
Therefore, the 19th term of the arithmetic progression is 25.
To find the sum of the 28th term, we can use the formula for the sum of an arithmetic sequence.
Step 3: Find the sum of the 28th term (S₂₈)
Sₙ = (n/2) * (a₁ + aₙ)
S₂₈ = (28/2) * (a₁ + a₂₈)
Substituting the values:
S₂₈ = 14 * (7 + (7 + (28 - 1) * 1))
S₂₈ = 14 * (7 + (35))
S₂₈ = 14 * (42)
S₂₈ = 588
Therefore, the sum of the 28th term of the arithmetic progression is 588.
a10 = 2*a2
7+9d = 2(7+d)
Solve to find d, and then
(A) a19 = a + 18d
(B) S28 = 28/2 (2*7 + 27d)