Question

the 12th term of an arithmetic sequence is 29. if the 3rd term is subtracted from the 7th term the result is -8. what is the 52nd term?​

Answer 1

Answer:

52th term of the A.P. is 109

Explanation:

We know that,

nth term of an A.P. :-

$\rm \implies \: a_n = a + (n - 1)d$

Where,

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

Now,

12th term of the A.P. series is 29. (given)

$\rm \therefore \: a_{12} = 29$

$\rm \implies \: a + (12 - 1)d = 29$

$\rm \implies \: a + 11d = 29$

$\rm \implies \: a = 29 - 11d$

Now,

it is given that, the difference between $\rm a_3$ and $\rm a_7$ is -8.

$\rm \therefore \: a_{3} - a_{7} = -8$

${ \rm \implies \{a + (3 - 1)d \} - \{a + (7 - 1)d \} = -8}$

$\rm \implies \{a + 2d \} - \{a + 6d \} = -8$

Substitute ‘a’ :-

${ \rm \implies \{29 - 11d + 2d \} - \{29 - 11d + 6d \} = - 8}$

Soliving further,

${ \rm \implies \{29 - 9d \} - \{29 - 5d \} = - 8}$

${ \rm \implies 29 - 9d - 29 + 5d = - 8}$

${ \rm \implies - 4d = - 8}$

${ \rm \therefore d = 2}$

Calculate the value of ‘a‘ :-

$\rm \implies \: a = 29 - 11d$

we have,

• d = 2

$\rm \implies \: a = 29 - 11(2)$

$\rm \implies \: a = 29 - 22$

$\rm \implies \: a = 7$

Now, 52th term of the A.P. is given by,

$\rm \implies \: a_{52} = 7 + (52 - 1)2$

$\rm \implies \: a_{52} = 7 + (51)2$

$\rm \implies \: a_{52} = 7 + 102$

$\rm \therefore \: a_{52} = 109$

∴ Required answer: 109

Answer 2

Answer:

the 12th term is −46

Step-by-step explanation:

A term of an arithmetic sequence can be calculated with the formula:

tn=a+(n−1)d

where:tn=any term in the arithmetic sequence a=  first term n=term number/number of terms, d= common difference

To find the 12th term of the sequence, we first need to find d, the common difference. We can do this by subtracting t1 from t2: t2−t1=14−20=−6

Now that you have the common difference, substitute all your known values into the formula to solve for

t12:

tn=a+(n−1)dt12=20+(12−1)(−6)t12=

20+(11)(−6)t12=

20−66t12=

−46

∴, the 12th term is −46.