The 13th termof an arithmetic sequence is 61 and the 20th term of the squence is 82. Calculate the sum of the first 25 terms of the sequence
T13 = a+12d = 61
T20 = a+19d = 82
so,
T20-T13 = 7d = 21
d = 3
a = 25
S25 = 25/2 (25 + 25+24*3)
= 25/2 (122) = 25*61 = 1525
To solve this problem, we need to first find the common difference (d) of the arithmetic sequence using the given information.
Let's call the 13th term of the sequence A₁₃ and the 20th term A₂₀.
We can use the formula for the nth term of an arithmetic sequence to find A₁₃ and A₂₀:
Aₙ = A₁ + (n - 1) × d
For A₁₃, we have:
61 = A₁ + (13 - 1) × d
61 = A₁ + 12d
For A₂₀, we have:
82 = A₁ + (20 - 1) × d
82 = A₁ + 19d
Now, we can solve these two equations simultaneously to find the values of A₁ and d.
Subtracting the first equation from the second equation, we get:
82 - 61 = (A₁ + 19d) - (A₁ + 12d)
21 = 7d
Dividing both sides by 7, we find:
d = 3
Now we can substitute the value of d back into either of the initial equations to find A₁. Let's use the equation for A₁₃:
61 = A₁ + 12(3)
61 = A₁ + 36
A₁ = 61 - 36
A₁ = 25
So, the first term of the sequence (A₁) is 25, and the common difference (d) is 3.
To calculate the sum of the first 25 terms of an arithmetic sequence, we can use the formula:
Sn = (n/2) × (A₁ + An)
We know that n = 25, A₁ = 25, and we need to find the 25th term (An).
To find An, we can use the formula for the nth term:
An = A₁ + (n - 1) × d
Substituting n = 25, A₁ = 25, and d = 3, we get:
A₂₅ = 25 + (25 - 1) × 3
A₂₅ = 25 + 24 × 3
A₂₅ = 25 + 72
A₂₅ = 97
Now we can calculate the sum:
Sn = (25/2) × (25 + 97)
Sn = 12.5 × 122
Sn = 1525
Therefore, the sum of the first 25 terms of the sequence is 1525.