A 8th term of an A.P is 380 and the 12th term is 58 ,find the 30th term
To find the 30th term of an arithmetic progression (A.P.), we need to find the common difference (d) first.
Given that the 8th term is 380 and the 12th term is 58, we can form two equations:
Eighth term: a + 7d = 380 (since the 8th term is a + 7d)
Twelfth term: a + 11d = 58 (since the 12th term is a + 11d)
To solve these two equations, we can subtract the second equation from the first equation:
(a + 7d) - (a + 11d) = 380 - 58
a + 7d - a - 11d = 322
-4d = 322
d = -322/4
d = -80.5
Now that we have the common difference, we can find the first term (a). We can substitute the value of d into one of the equations:
a + 11d = 58
a + 11(-80.5) = 58
a - 885.5 = 58
a = 58 + 885.5
a = 943.5
So, the first term (a) is 943.5 and the common difference (d) is -80.5.
Now, we can find the 30th term by using the formula for the nth term of an A.P:
nth term = a + (n - 1)d
Substituting the values:
30th term = 943.5 + (30 - 1) * -80.5
= 943.5 + 29 * -80.5
= 943.5 - 2334.5
= -1391
Therefore, the 30th term of the A.P. is -1391.
To find the 30th term of an arithmetic progression (A.P.), we need to first determine the common difference (d).
Given:
8th term (a₈) = 380
12th term (a₁₂) = 58
Using the formula for the nth term of an A.P.: aₙ = a₁ + (n - 1)d
Substitute a₈ = 380:
a₈ = a₁ + (8 - 1)d
380 = a₁ + 7d .......(1)
Substitute a₁₂ = 58:
a₁₂ = a₁ + (12 - 1)d
58 = a₁ + 11d .......(2)
Now, we have a system of two equations:
a₁ + 7d = 380 .......(1)
a₁ + 11d = 58 .......(2)
Subtract equation (2) from equation (1) to eliminate a₁:
a₁ + 7d - (a₁ + 11d) = 380 - 58
-4d = 322
d = -322/4
d = -80.5
Substitute the value of d back into equation (1) to find a₁:
a₁ + 7(-80.5) = 380
a₁ - 563.5 = 380
a₁ = 380 + 563.5
a₁ = 943.5
Now that we know the first term (a₁ = 943.5) and the common difference (d = -80.5), we can find the 30th term (a₃₀) using the formula:
a₃₀ = a₁ + (30 - 1)d
a₃₀ = 943.5 + 29*(-80.5)
a₃₀ = 943.5 - 2334.5
a₃₀ = -1391
Therefore, the 30th term of the arithmetic progression is -1391.
To find the 30th term of an arithmetic progression (AP), we can use the formula:
an = a1 + (n-1)d
where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.
Given that the 8th term is 380 and the 12th term is 58, we can write two equations using the above formula:
a8 = a1 + (8-1)d
a12 = a1 + (12-1)d
Using these equations, we can solve for a1 and d.
a8 = a1 + 7d (equation 1)
a12 = a1 + 11d (equation 2)
Subtracting equation 1 from equation 2, we have:
a12 - a8 = (a1 + 11d) - (a1 + 7d)
58 - 380 = 4d
-322 = 4d
d = -322/4
d = -80.5
Now, substitute the value of d into equation 1 to solve for a1:
a8 = a1 + 7(-80.5)
380 = a1 - 563.5
a1 = 380 + 563.5
a1 = 943.5
We now have the values of a1 and d, so we can find the 30th term using the formula:
a30 = a1 + (30-1)d
a30 = 943.5 + 29(-80.5)
a30 = 943.5 - 2339.5
a30 = -1396
Therefore, the 30th term of the arithmetic progression is -1396.