Question

Find the sum of first 24 terms of the list of the numbers whose nth term is given by an = 3 + 2n

Answer 1

Solution :

Given :

$\\ \large \sf \: nth \: \: term = 3 + 2n \\ \\ \large \sf \: sum \: \: of \: 24 \: terms = {?} \\ \\ \large \sf \underline{finding : } \\ \\ \large \sf \: taking \: \: (n = 1) \\ \\ \large \sf : \implies \: a1 = 3 + 2 \times 1\\ \\ \large \sf : \implies \: a1 = 5 \\ \\ \large \sf \: taking \: \: (n = 2) \\ \\ \large \sf : \implies \: a2 = 3 + 2 \times 2 \\ \\ \large \sf : \implies \: a2 = 7$

Now,

$\\ \\ \large \sf \: common \: \: difference = a2 - a1 \\ \\ \large \sf \implies \: d = 7 - 5 = 2$

Now..

$\\ \\ \large \sf \: sum \: \: of \: \: nth \: \: term : \\ \\ \large \tt : \implies \frac{n}{2} [2a + (n - 1)d] \\ \\ \\ \large \tt : \implies \frac{ \cancel{24}}{ \cancel2} [2 \times 5+ (24- 1)2] \\ \\ \\ \large \tt : \implies 12 [10 + 23 \times 2] \\ \\ \\ \large \tt : \implies 12 \times 56$

Hence, Sum of 24th term of the given A.P is 672

Answer 2

Given :-

• an = 3 + 2n

To find :-

• sum of Frist 24 terms

Solution :-

$\sf\boxed{\bold{\pink{1st \: term : - }}}$

$\tt = a1 \: = 3 + 2(1) \\ \\ \tt = a \: = 3 + 2 \\ \\ \tt \: = a \: = 5$

$\sf\boxed{\bold{\pink{2nd \: term: - }}}$

$\tt = a2 = 3 + 2(2) \\ \\ \tt = a \: = 3 + 4 \\ \\ \tt = a = 7$

$\sf\boxed{\bold{\pink{3rd \: term : - \: }}}$

$\tt =a3 = 3 + 2(3) \\ \\ \tt = a = 3 + 6 \\ \\ \tt = a \: = 9$

$\sf\boxed{\bold{\pink{4th \: term : - }}}$

$\tt = a4 = 3 + 2(4) \\ \\ \tt = a = 3 + 8 \\ \\ \tt = a = 11$

so therefore the series are 5,7,9,11

so therefore difference between consecutive sums are same

so therefore it is an ap

now we need to find the sum of Frist 24 terms

now,

n = 24

a = 5

d = 7 - 5 = 2

now by putting values in the formula :-

$:\implies\sf \: sum \: = \frac{n}{2} (2a + (n - 1)d) \\ \\ \\ :\implies\sf \: \frac{24}{2} (2 \times 5 + (24 - 1)2) \\ \\ \\ :\implies\sf \: 12(10 + (23)2)$

$:\implies\sf \: 12(10 + 46) \\ \\ \\ :\implies\sf \: 12 \times 56 \\ \\ \\ :\implies\sf \: 672$

$\sf\boxed{\bold{\pink{so \: therefore \: the \: sum \: of \: 24 \: terms \: are \: 672}}}$