sum of the first 4 terms of an arithmetic sequence is 72. sum of the first 9 terms is also 72. what is the 5th term of the arithmetic sequence?
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SHONEJOSEPH1715D
3 weeks ago
Math
Secondary School
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Sum of first 4 terms and the sum of first 9 terms of an arithmetic sequence are72.Write the fifth term.
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ANSWER:
The fifth term = 8.
GIVEN:
Sum of first 4 terms and the sum of first 9 terms of an A.P are 72.
TO FIND:
The fifth term.
EXPLANATION:
Sₙ = n/2(2a + (n - 1)d)
S₄ = S₉ = 72
S₄ = 4/2(2a + (4 - 1)d) = 72
2(2a + 3d) = 72
2a + 3d = 36 --------> 1
S₉ = 9/2(2a + (9 - 1)d) = 72
(2a + 8d) = 8 × 2
2a + 8d = 16 -------> 2
Subtract the two equations
2a + 3d - 2a - 8d = 36 - 16
- 5d = 20
d = - 4
Substitute d = - 4 in equation 1
2a + 3( - 4) = 36
2a - 12 = 36
2a = 48
a = 24
Tₙ = a + (n - 1)d
T₅ = 24 + (5 - 1)( - 4)
T₅ = 24 + 4(- 4)
T₅ = 24 - 16
T₅ = 8
HENCE THE FIFTH TERM = 8.
VERIFICATION:
S₄ = 4/2(2a + (4 - 1)d) = 72
S₄ = 2(2a + 3d)
Substitute a = 24 and d = - 4
S₄ = 2[2(24) + 3(- 4)]
S₄ = 2(48 - 12)
S₄ = 2(36)
S₄ = 72
S₉ = 9/2(2a + (9 - 1)d) = 72
S₉ = 9/2(2a + 8d)
Substitute a = 24 and d = - 4
S₉ = 9/2[2(24) + 8(- 4)]
S₉ = 9/2(48 - 32)
S₉ = 9/2(16)
S₉ = 9 × 8
S₉ = 72
S₄ = S₉ = 72