find the first 3 term s of an arithmetic sequence in which the 21 term is -54 and the fifteenth term is -170
given:
t(21) = a+20d = -54
t(15) = a+14d = -170
subtract them
6d = 116
d = 58/3
back into the 1st equation
a+20(58/3) = -54
a = -1322/3
so the first 3 terms are -1322/3, -1264/3, -402
To find the first three terms of an arithmetic sequence, we need to find the common difference (d) and the first term (a₁).
Given that the 21st term (a₂₁) is -54 and the 15th term (a₁₅) is -170, we can use these two values to find the common difference.
We can start by finding the difference between the 15th term and the 21st term:
-170 - (-54) = -170 + 54 = -116
The difference between the 15th term and the 21st term is -116.
Since the nth term of an arithmetic sequence can be represented as aₙ = a₁ + (n - 1)d, we can use the 15th term to find the value of a₁:
-170 = a₁ + (15-1)d
-170 = a₁ + 14d
Similarly, we can use the 21st term to find the value of a₁:
-54 = a₁ + (21-1)d
-54 = a₁ + 20d
We now have two equations with two unknowns (a₁ and d). We can solve these equations simultaneously to find the values of a₁ and d.
Equation 1: -170 = a₁ + 14d
Equation 2: -54 = a₁ + 20d
To eliminate a₁, we can subtract Equation 2 from Equation 1:
-170 - (-54) = (a₁ + 14d) - (a₁ + 20d)
-170 + 54 = a₁ + 14d - a₁ - 20d
-116 = -6d
d = -116 / -6
d = 19.33 (rounded to two decimal places)
Now that we have the value of d, we can substitute it back into one of the equations to find the value of a₁:
-54 = a₁ + 20d
-54 = a₁ + 20 * 19.33
-54 = a₁ + 386.6
a₁ = -54 - 386.6
a₁ = -440.6 (rounded to one decimal place)
The first term (a₁) is approximately -440.6, and the common difference (d) is approximately 19.33.
To find the first three terms, we can use the formula aₙ = a₁ + (n - 1)d:
a₁ = -440.6
a₂ = -440.6 + (2 - 1) * 19.33
a₃ = -440.6 + (3 - 1) * 19.33
Calculating the values:
a₂ = -440.6 + 19.33
a₃ = -440.6 + 38.66
Rounding the values to two decimal places:
a₁ ≈ -440.6
a₂ ≈ -421.27
a₃ ≈ -401.94
Therefore, the first three terms of the arithmetic sequence are approximately -440.6, -421.27, and -401.94.