Question

if an ap the first term is 2 and sum of the first five terms is one fourth of the next five terms then 10th term is​

Answer 1

EXPLANATION.

→ First term of an Ap = 2.

→ Sum of the first five terms is one - fourth

of the next five terms.

→ To find 10th terms of an Ap.

$\sf \: nth \: terms \: of \: an \: ap \\ \\ \sf \: \: a_{n} = a + (n - 1)d \\ \\ \sf \: t_{1} + t_{2} + t_{3} + t_{4} + t_{5} = \frac{1}{4}( t_{6} + t_{7} + t_{8} + t_{9} + t_{10})$

$\sf \: t_{1} = a \\ \\ \sf : \implies \: \: t_{2} = a + d \\ \\ \sf \: t_{3} = a + 2d \\ \\ \sf \: \: t_{4} = a + 3d \\ \\ \sf : \implies \: t_{5} = a + 4d$

$\sf \: \: sum \: of \:first \: five \: terms \\ \\ \sf \: \: a + a + d + a + 2d + a + 3d + a + 4d \\ \\ \sf \:5a + 10d$

$\sf \: t_{6} = a + 5d \\ \\ \sf \: t_{7} = a + 6d \\ \\ \sf \: t_{8} = a + 7d \\ \\\sf \: t_{9} = a + 8d \\ \\ \sf \: t_{10} = a + 9d$

$\sf \:a + 5d + a + 6d + a + 7d + a + 8d + a + 9d \\ \\\sf \:5a + 35d$

$\sf \:(5a + 10d) = \dfrac{1}{4}(5a + 35d) \\ \\ \sf \:4(5a + 10d) = 5a + 35d \\ \\ \sf \: \: 20a + 40d = 5a + 35d \\ \\ \sf \:15a = - 5d \\ \\ \sf \:15(2) = - 5d \\ \\ \sf \:30 = - 5d \\ \\ \sf \:d = - 6$

$\sf : \implies \: t_{10} = a + 9d \\ \\ \sf : \implies \:2 + 9( - 6) \\ \\ \sf : \implies \:2 - 54 = - 52$

Answer 2

Step-by-step explanation:

Given :

• if an ap the first term is 2 and sum the next five terms

• the sum of first five terms is one-fourth of the next five terms,

To Find :

• Find the 10th terms

Solution :

[a + (a + d) + (a + 2d) + (a + 3d) + (a + 4d)] = 1/4[(a + 5d) + (a + 6d) + (a + 7d) + (a + 8d) + (a + 9d)]

(5a + 10d) = 1/4(5a + 35d)

20a + 40d = 5a + 35d

15a + 5d = 0

3a + d = 0

Substitute the value of a

D = -3 × 2

D = - 6

The 10th term = a + (10 - 1)d

= a + 9d

= 2 + (9 × - 6)

= 2 - 54

= - 52

The 10th term of the given AP is - 52