Question

if 4thterm of an ap is 14 and its 12 th term is 70 then what is the 1st term​

Answer 1

Question :-

If $\rm{4 {}^{th} }$ term of an AP is 14 and it's $\rm{12 {}^{th} }$ is 70, then what is the 1st term.

Solution :-

C A S E (I) :

If 4th term if 14.

As we can write,

$\sf:\longmapsto{a + 3d = 14 - - - - - (1)}$

C A S E (II) :

If 12th term is 70.

Also we can write,

$\sf:\longmapsto{a + 11d = 70 - - - - (2)}$

Now, Subtract Eqⁿ (1) from (2)

$\sf:\longmapsto{a + 11d - (a + 3d) = 70 - 14} \\ \\ \sf:\longmapsto{a - a + 11d - 3d = 56} \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{8d = 56} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \sf:\longmapsto{d = \frac{56}{8} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \bf:\longmapsto \red{d = 7} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:$

Substituting d = 7 in Eqⁿ (1)

$\sf:\longmapsto{a + 3(7) = 14} \\ \\ \sf:\longmapsto{a + 21 = 14} \: \: \\ \\ \bf:\longmapsto \red{a = - 7} \: \: \: \: \: \: \: \:$

We know that,

• a = first term.

Hence,

• 1st term is -7.

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