Question

If the 7th term of an A.P is 14 and 12 term is 24. Find the first term and the common difference.

Answer 1

to go from the 7th term, to the 12th term in an arithmetic sequence, you'd need to use the common difference "d", 5 times, thus


`\bf \begin{array}{cll} term&value\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ 7&a_7=14\\ 8&a_8=14+d\\ 9&a_9=14+d+d\\ 10&a_(10)=14+d+d+d\\ 11&a_(11)=14+d+d+d+d\\ 12&a_(12)=14+d+d+d+d+d\\ &a_(12)=14+5d\\ &a_(12)=24 \end{array}\implies \stackrel{a_(12)}{24}=\stackrel{a_(12)}{14+5d} \\\\\\ 10=5d\implies \cfrac{10}{5}=d\implies \boxed{2=d}\\\\ -------------------------------\\\\`


`\bf n^(th)\textit{ term of an arithmetic sequence}\\\\ a_n=a_1+(n-1)d\qquad \begin{cases} n=n^(th)\ term\\ a_1=\textit{first term's value}\\ d=\textit{common difference}\\ ----------\\ d=2\\ n=7\\ a_7=14 \end{cases} \\\\\\ a_7=a_1+(7-1)2\implies 14=a_1+(7-1)2\implies 14=a_1+12 \\\\\\ 14-12=a_1\implies \boxed{2=a_1}`

Answer 2

Information provided with us :

• 7th term of the A.P. is 14.
• 12th term of the A.P. is 24.

What we have to calculate :

• First term (a) and common difference (d) of the A.P.?

Performing Calculations :

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

• tn = a + (n - 1) d

Here in this formula,

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

For 7th term of the A.P. :

=> 14 = a + (7 - 1) d

=> 14 = a + 6d

=> a = 14 - 6d

For 12th term of the A.P. :

=> 24 = a + (12 - 1) d

=> 24 = a + 11 × d

=> 24 = a + 11d

Substitute the value of a which we got above,

=> 24 = 14 - 6d + 11d

=> 24 = 14 + 5d

=> 5d = 24 - 14

=> 5d = 10

=> d = 2

★ Therefore, common difference is 2..!!

Finding out first term (a) of the A.P. :

=> a = 14 - 6d

=> a = 14 - 6 (2)

=> a = 14 - 6 × 2

=> a = 14 - 12

=> a = 2

★ Therefore, first term of the A.P. is 2..!!