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Sums based on ‘a_n’ Formula

ARITHMETIC

PROGRESSIONS

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a₁₆ = 73

7) Find the 31^st term of an AP whose 11^th term is 38 and

the 16^th term is 73:

Sol:

For given AP:

a₁₁ = 38,

a₁₁ = a + 10d

∴ 38

= a + 10d

…..(i)

a₁₆ = a + 15d

∴ 73

= a + 15d

…..(ii)

Subtracting (i) from (ii)

a + 15d =

73

a + 10d =

38

( - )

5d =

35

d = 7

Substituting d = 7 in (i)

a + 10(7) =

38

∴ a + 70 =

38

∴ a =

38 – 70

∴ a =

– 32

Now, lets find a₃₁

a₃₁ = a + 30d

= – 32 + 30(7)

∴ a₃₁ = 178

= – 32 + 210

∴ 31^st term of AP is 178.

We know that,

We need to find a₃₁

These are linear eqⁿ

in 2 variables a & d

Since, terms with variable ‘a’

has same coefficient and same sign

∴Subtract the two equations

Exercise 5.2 7

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8) An AP consists of 50 terms of which 3^rd term is 12 and

last term is 106. Find the 29^th term.

Sol:

We need to find a₂₉

For given AP:

a₃ = 12,

a₅₀ = 106

a₃ = a + 2d

∴ 12

= a + 2d

…..(i)

a₅₀ = a + 49d

∴ 106

= a + 49d

…..(ii)

Subtracting (i) from (ii)

a + 49d =

106

a + 2d =

12

( - )

47d =

94

d = 2

These are linear eqⁿ

in 2 variables a & d

Substituting d = 2 in (i)

a + 2(2) =

12

∴ a + 4 =

12

∴ a =

12 – 4

∴ a =

8

Now, lets find a₂₉

a₂₉ = a + 28d

= 8 + 28(2)

∴ a₂₉ = 64

= 8 + 56

∴ 29^th term of AP is 64.

We know that,

Since, AP consist of 50 terms

Then, last term is a₅₀

Since, terms with variable ‘a’

has same coefficient and same sign

Subtract the two equations

Exercise 5.2 8

HOMEWORK