Question

Sum of first 4 terms and the sum of first 9 terms of an arithmetic sequence are72.Write the fifth term.

Answer 1

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

HENCE VERIFIED.