Question

Find the first terms and common difference of an A.P. whose t₈ = 3 and t₁₂ = 52.

Answer 1

For an A.P., let a be the first term and d be the common difference.

t₈ = 3 and t₁₂ = 52    ......[Given]

Since t_n = a + (n – 1)d,

t₈ = a + (8 – 1)d

∴ 3 = a + 7d

i.e., a + 7d = 3      ......(i)

Also, t₁₂ = 52

∴ a + (12 – 1)d = 52

∴ a + 11d = 52   ......(ii)

Subtracting equation (i) from (ii), we get

a + 11d = 52
a +   7d = 3
–      –       –    
        4d = 49

∴ d = `49/4`

Substituting d = `49/4` in equation (i), we get

`"a" + 7(49/4)` = 3

∴ `"a" + 343/4` = 3

∴ a = `3 - 343/4`

= `(-331)/4`

∴ The first term and common difference of A.P. are `(-331)/4` and `49/4` respectively.