Question

Find the sum of the following without adding with explaination:-
7+13+19+......+43?

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Answer 1

Required Answer:-

To FinD:-

• 7 + 13 + 19 +.......+ 43 without actually adding the numbers directly$.$

Step-wise-Step Explanation:

As we can that the numbers follow a definite sequence i.e. they have a common difference of 13 - 7 = 19 - 13 = ...... = 6. Hence it is a arithmetic progression.

Determining the no. of terms:

• l = a + (n - 1)d

Where,

• l = Last term of the AP
• a = First term of the AP
• n = no. of terms
• d = common difference

Plugging the values:

➛ 43 = 7 + (n - 1)6

➛ 36 = 6(n - 1)

➛ n - 1 = 6

➛ n = 7

Sum of 7 terms of the AP:

• Sn = n/2 { a + l }

Where,

• Sn = Sum of n terms.
• n = no. of terms
• a = First term
• l = last term

Plugging the values:

➛ S7 = 7/2 { 7 + 43 }

➛ S7 = 7/2 { 50}

➛ S7 = 7 × 25

➛ S7 = 175

Hence:-

The required sum of the numbers in the sequence is 175 (Answer).

Answer 2

Answer:

$\huge \bf \: To \: find$

Sum without addition

$\huge \bf \: Solution$

Do you know ? The series are following a pattern.

$19 - 6 = 13$

$13 - 6 = 7$

Now,

• L = a + (n-1) d

L = Last term of AP

A = First term of AP

N = No. of terms

D = Common difference

$\sf \: 43 = 7 + (n - 1)6$

$\sf \: 36 = 6(n - 1)$

$\sf \dfrac{36}{6} = n - 1$

$\sf \: 6 = n + 1$

$\sf \: n \: = 6 + 1$

$\sf \: n \: = 7$

Total terms = 7

NOW,

Sn = n/2 { a + l }

Where,

Sn = Sum of n terms.

n = no. of terms

a = First term

l = last term

$\sf \: S7 = \dfrac{7}{2}$

$\sf \: S7 = \dfrac{7}{2} \: ({7 + 43})$

$\sf \: S7 = \dfrac{7}{2} \times 50$

$\sf \: S7 = 7 \times 25$

$\huge \bf \: S7 = 175 \bigstar$