Question

The 10th term of an a.p is -27 and the 5th term is -12 .find the first term?​

Answer 1

Explanation:

`a_(10) = - 27... (Given) \\\\</p><p>\therefore a + 9d = - 27...... (1)\\\\</p><p></p><p>a_(5) = - 12... (Given) \\\\</p><p>\therefore a + 4d = - 12.......(2)\\\\</p><p>`

Subtracting equation (2) from equation (1)

`a + 9d-( a + 4d )= - 27-(-12)\\\\</p><p>\therefore a +9d - a-4d = - 27+12\\\\</p><p>\therefore 5d = - 15\\\\</p><p>\therefore d = \frac {- 15}{5}\\\\</p><p>\huge \orange {\boxed {\therefore d = -3}} \\\\</p><p>`

Substituting d = - 3 in equation (1), we find:

`a + 9* (-3)= - 27\\\\</p><p>\therefore a - 27 = - 27\\\\</p><p>\therefore a = 27- 27\\\\</p><p>\huge \red {\boxed {\therefore a = 0}}\\\\`

Hence, first term is zero.

Answer 2

Information provided with us :

• 10th term of an A.P. is -27
• 5th term is -12

What we have to calculate :

• First term of the A.P. ?

Performing Calculations :

Now, as we clearly know that nth or general term of an A.P. (Arithmetic progression) s calculated by the formula :

• tn = a + (n - 1) d

Here in this formula,

• a denotes first term
• d is common difference
• n is number of terms

★ For 10th term of the A.P. :

=> a + (10 - 1) d = -27

=> a + (9) d = -27

=> a + 9 × d = -27

=> a + 9d = -27

=> a = -27 - 9d

Here we got a temporary value of first term.

★ For 5th term of the A.P. :

=> a + (5 - 1) d = -12

=> a + 4d = -12

Substituting the value of a which we got above,

=> -27 - 9d + 4d = -12

=> -27 - 5d = -12

=> -5d = -12 + 27

=> -5d = 15

=> -d = 15 / 5

=> -d = 3

=> d = -3

Therefore, common difference (d) is -3.

★ Finding out first term of A.P. :

=> a = -27 - 9d

=> a = -27 - 9 (-3)

=> a = -27 - 9 × -3

=> a = -27 - 27

=> a = 0

★ Henceforth,

• First term of the A.P. is 0