Question

ارا
If the sum of the first 8term of AP 136and last of first 15 terms
is 465 then find the sum of first 25 terms​

Answer 1

Answer:

The sum of first 25 terms is 1321.

Formula used:

$\sf{t_{n} = a + (n - 1)d}$

Step-by-step explanation:

Given that,

• $\sf{t_8}$ = 136
• $\sf{t_{15}}$ = 465

Case (I),

$\red\bigstar$ $\sf{t_8}$ = 136

▶ $\sf{t_n}$ = a + (n - 1)d

▶ $\sf{t_8}$ = a + (8 - 1)d

▶ 136 = a + 7d _________________eqn. (1)

Case (II),

$\green\bigstar$$\sf{t_{15}}$ = 465

▶ $\sf{t_n}$ = a + (n - 1)d

▶ $\sf{t_15}$ = a + (15 - 1)d

▶ 465 = a + 14d________________eqn. (2)

From eqn (1) & (2),

$\sf{\cancel{a }+ 7d = 136} \\ </p><p></p><p>\sf{ \cancel{a} + 14d = 465 } \\ - - - - - - \\ \: \: \: \: \: \: \: \: \sf {7d = 329}\\ \\ \longrightarrow{\pink{\sf{d = 47}}}$

Put d = 47 in eqn. (1) , We get :

▶ a + 7(47) = 136

▶ a + 329 = 136

▶ a = 329 - 136

▶ a = 193

$\longrightarrow{\sf{\blue{ a = 193}}}$

Case (III),

$\sf{t_{25} = ? }$

▶$\sf{t_{n}}$ = a + (n - 1)d

▶$\sf{t_{25}}$ = 193 + (25-1) × 47

▶$\sf{t_{25}}$ = 193 + 24 × 47

▶$\sf{t_{25}}$ = 193 + 1128

$\blue\bigstar$ ${\red{\sf{t_{25} = 1321}}}$